de Sitter Space as a Glauber-Sudarshan State
PPreprint typeset in JHEP style - HYPER VERSION de Sitter Space as a Glauber-Sudarshan State
Suddhasattwa Brahma , Keshav Dasgupta , Radu Tatar Department of Physics, McGill University, Montr´eal, Qu´ebec, H3A 2T8, Canada Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 7ZL,United Kingdom [email protected]@hep.physics.mcgill.ca, [email protected]
Abstract:
Glauber-Sudarshan states, sometimes simply referred to as Glauber states, oralternatively as coherent and squeezed-coherent states, are interesting states in the config-uration spaces of any quantum field theories, that closely resemble classical trajectories inspace-time. In this paper, we identify four-dimensional de Sitter space as a coherent stateover a supersymmetric Minkowski vacuum. Although such an identification is not new,what is new however is the claim that this is realizable in full string theory, but only inconjunction with temporally varying degrees of freedom and quantum corrections resultingfrom them. Furthermore, fluctuations over the de Sitter space is governed by a generalizedgraviton (and flux)-added coherent state, also known as the Agarwal-Tara state. The real-ization of de Sitter space as a state, and not as a vacuum, resolves many issues associatedwith its entropy, zero-point energy and trans-Planckian censorship, amongst other things. a r X i v : . [ h e p - t h ] S e p ontents
1. Introduction and summary 1
2. de Sitter space as a Glauber-Sudarshan state 14
3. Quantum effects and the Schwinger-Dyson equations 53
4. Properties of the Glauber-Sudarshan state 97
5. Discussions and conclusions 111
1. Introduction and summary
The path towards an actual realization of a four-dimensional de Sitter vacuum in stringtheory is long and arduous with obstacles coming, for example, from the no-go conditionswith increasing sophistications. The original no-go condition, given by Gibbons [1], ruledout the possibility of realizing four-dimensional de Sitter vacuum with supergravity fluxes.This was followed by a more refined version from Maldacena and Nunez [2], that excluded– 1 –ranes and anti-branes. The conclusion was that neither branes nor anti-branes can providepositive cosmological constant solutions, although anti-branes could break supersymmetryspontaneously. More recently, however, it was proposed in [3] that even the O-planes cannot help, so eventually the sole burden of realizing four-dimensional de Sitter vacua in stringtheory rested on the shoulder of the quantum terms. Unfortunately, as also shown in [3, 4],this is easier said than done: the quantum terms are more delicate as they are constrainedby a condition that seemed to be satisfied only if there existed an inherent hierarchy . Inthe analysis of [3], which was done from M-theory point of view, the hierarchy could bebetween g s , the type IIA string coupling, and M p , the Planck scale. Generically, as in anyquantum system, one would expect that at least the M p hierarchy could be easily attainedas supergravity operators typically have different M p scalings. All in all, the belief tillearly 2018 was that, quantum corrections could in principle do the job. Subtleties like fine-tunings etc should go hand in hand, but at the deepest level, there would be no furtherobstructions that could prohibit four-dimensional de Sitter vacua to exist in string theory.With a little stretch, solutions like [5], should then be part of the ensemble of models thatdemonstrate the existence of stringy de Sitter vacua.Such a rosy picture did not last long as objections were raised from many directions.Early objections, specifically regarding [5], were either aimed mostly at the usage of anti-D3 branes [6, 7], or at the choice of the supersymmetric AdS vacua [8], in the constructionsof stringy IIB de Sitter vacua. The former objections were shown to not exist if anti-D3branes (in the presence of O3-planes) broke supersymmetry spontaneously [9], and, in fact,in [10, 11] numerous constructions of de Sitter vacua in IIA were shown using anti-branes(plus O-planes and quantum corrections [10]) as the primary ingredients.The latter objections, primarily from [8], were more subtle. It questioned the funda-mental edifice on which the whole construction of [5] rested, namely the four-dimensionalsupersymmetric AdS vacua. The claim in [8] was that an AdS vacuum simply could not bea good starting point for uplifting to a non-supersymmetric de Sitter vacuum. Subtletiescome from the quantum terms, mostly involving the quartic order curvature terms, thattypically lead to rolling solutions. Thus backgrounds involving no AdS vacua, somewhatalong the lines of [12], might be more productive.The situation by mid-2018 was then an atmosphere of hopeful optimism. While theobjections against the existence of four-dimensional de Sitter vacua from string theorywere acknowledged [13], the way out of these did not seem hard either. Again with a littleextra fine tunings, and some minor adjustments, the construction of [5] still seemed to holdwater.A more severe blow to this construction, and in particular to any constructions thataimed to reproduce a positive cosmological solution from string theory, appeared from aseries of papers starting roughly from mid-June of 2018 [14, 15, 16]. These set of paperschallenged the very existence of a positive cosmological constant solution in a consistenttheory of quantum gravity citing contradictions coming from the weak gravity conjecture[17] as well as from bounds on derivatives of the potential on the landscape [14] and otherrelated issues. This was followed by a slew of papers, a subset of which is cited in [18] (see[19] for reviews), that either provided support or challenged these ideas. The latter mainly– 2 –uestioned the ad hoc nature of the conjectures in [14, 15, 16] and pointed out issues wherethe derivative bounds were too stringent to match with the experimental values. Questionswere also raised, for example in [20], against the non-existence of effective field theory(EFT) descriptions propagated by [14, 15, 16], although in retrospect it is now clear from[4, 21, 22, 23] that in the absence of time-dependent degrees of freedom, a four-dimensionalEFT description fails to arise. This does lend some credence to the swampland conjecturesof [14, 15, 16], but the works of [4, 21, 22, 23] approached the issue of the existence of anEFT following a very different path by actually analyzing the infinite class of interactions,both local and non-local, in a systematic way. In this paper, and especially in sections 3.1and 3.2, we will analyze the non-perturbative effects from instantons and world-volumefermions that were not discussed in [4, 21, 22, 23]. In the absence of time-dependentdegrees of freedom, all our computations lead to one conclusion: the loss of both g s andM p hierarchies from the quantum loops, implying a severe breakdown of an EFT descriptionin four-dimensions.The ad hoc nature of the conjectures in [14, 15, 16] were soon given a slightly bettertheoretical motivation in [24]. The aim of [24] was basically to blend the trans-Planckianissue raised in [25] with the swampland conjectures of [14, 15, 16], and rechristen them in apackaged form as the Trans-Planckian Censorship Conjecture (TCC). The trans-Planckianissue of cosmology, especially in the presence of accelerating backgrounds (de Sitter spacebeing the prime example), challenges the very notion of Wilsonian effective action becauseany fluctuations over these backgrounds create modes that have time- dependent frequencies.Since frequencies are related to the energies, and Wilsonian method involve integrating out the high energy modes, we face a severe conundrum: how to integrate out high energymodes when the energies themselves are changing with respect to time? One way wouldbe to integrate out the modes at every instant of time. This is not a very efficient way,because the integrated out UV modes will become IR at a later time, leading to a waterfall like structure where the depleted IR modes get continually replaced by the red-shifted UVmodes. Additionally, in a theory of gravity the far UV modes, i.e. the modes beyond M p ,are not well defined because there is no UV completion like we have in string theory. Thesemodes will show up at a later time, thus bringing in the UV issue now at low energies, unlesswe have a way to censor these modes. Such censoring will rule out inflation, or at leastsufficiently long-lasting inflation to avoid fine-tuning issues, along with all the advantagesthat we have assimilated through inflationary dynamics.The trans-Planckian problem [25] or it’s new guise, the TCC [24], relies on two things:one, the time-dependent frequencies in an accelerating background, and two, the far UVmodes that do not have well-defined dynamics ( i.e. they could even be non-unitary).However if we view string theory as the UV completion of gravity (and other interactions),there appears no basis to the second point as the UV modes clearly have well-definedunitary dynamics! The first point however is still a concern: the modes do change withrespect to time, so Wilsonian method will still be hard to perform in any acceleratingbackground. Is there a way out of this?It turns out, if we view de Sitter space to be a state instead of a vacuum , then thereis a way out. Such a state should, at best, replace the classical configuration to something– 3 –hat closely resembles it. This means what we are looking for is a Glauber-Sudarshan state[26, 27], alternatively, known as a coherent state [28]. This identification is not entirelynew as the first construction of de Sitter space as a coherent state in four-dimensionalquantum gravity has appeared in [29] (see also [30]), but what is new here is the claim thatsuch a state may be realized in full string theory and is quantum mechanically stable . Thestability is crucial as there are literally an infinite number of possible local and non-local,including their perturbative, non-perturbative and topological, quantum corrections. Oneof our aims here is to justify the stability of the Glauber-Sudarshan state amidst all thesecorrections.Viewing de Sitter space as a state instead of a vacuum resolves other issues relatedto zero point energy, supersymmetry breaking, entropy etc that we will concentrate onhere. The first two are easy to understand. The zero point energies from the bosonic andthe fermionic degrees of freedom cancel once we take a supersymmetric warped-Minkowskibackground. Once supersymmetry is broken spontaneously by the Glauber-Sudarshanstate, the cosmological constant Λ is determined from the fluxes and the quantum correc-tions, with no contributions from the zero-point energies [21]. The spontaneous breakingof supersymmetry arises from the expectation values of the fluxes − in this case G-fluxesin the M-theory uplift of the IIB model − over the eight-manifold when they are no longerself-dual. These fluxes are self-dual over the vacuum eight-manifold, so it’s the expectationvalues that break supersymmetry.The issue of entropy is bit more non-trivial, and we shall elaborate on this in section4.3. The important question here is the reason for a finite entropy of the de Sitter space,when a Minkowski space generically has an infinite entropy. Since we define our de Sitterspace as a Glauber-Sudarshan state over a warped-Minkowski background, should this notbe a concern now? The answer has to do with the interacting Hamiltonian, as well as thefiniteness of the number of gravitons in the Glauber-Sudarshan state defined over someHubble patch; including other criteria that we shall elaborate in section 4.3.There are also other related issues that enter the very definition of the interactingHamiltonian that is useful in resolving the entropy puzzle, and they have to do with theinfinite collections of perturbative and non-perturbative corrections. These correctionsappear as series, and so convergence of the series is important. We will discuss them insection 3.2. Getting de Sitter is a hard problem not just because of the no-go conditions [1, 2, 3], orbecause of the swampland constraints [14, 15, 16] − the former can be easily overcomeby taking quantum corrections, and the latter by switching on time-dependent degrees offreedom or by viewing de Sitter as a Glauber-Sudarshan state − but because of the fact thatthe analysis to show the existence of a de Sitter space, either as a state or by solving theSchwinger-Dyson’s equations (see details in section 3.3), is technically challenging. Why isthat so?The reason is not hard to see. Imagine we want to express (2.2), or more appropriatelythe M-theory uplifted background (2.4), as a consequence of the Glauber-Sudarshan state– 4 –n say M-theory (what this means will be elaborated later. Here we simply sketch thepicture). A background like (2.4) can exist if it solves some equations of motion (EOMs).These EOMs turn out be the Schwinger-Dyson’s equations, which are like the Ehrenfest’sequations in quantum mechanics, meaning that they appear as EOMs for expectation val-ues . Thus M-theory equations appear rather surprisingly as Schwinger-Dyson’s equations(details are in section 3.3). Such simple-minded statement entails some important conse-quences that have been largely ignored in the literature. In the following, we list some ofthese consequences.The first and the foremost of them is to solve the time-dependent EOMs, either assupergravity EOMs in the presence of all the quantum corrections mentioned above, or asa consequence of the Schwinger-Dyson’s equations. This, by itself, is challenging because,unless we have a way to control the infinite set of quantum terms, there is no simple wayto express them. The EOMs however can only provide a local picture, but the existenceof a solution, or even the Glauber-Sudarshan state, relies heavily on global constraintstoo. The global constraints come from flux quantizations, anomaly cancellations etc, andtherefore we have to (a) not only solve the time-dependent EOMs, but also (b) explainhow fluxes remain quantized with time-dependences, (c) how anomaly cancellations work,(d) how moduli stabilization may be understood when the moduli themselves are varyingwith time, (e) how the no-go conditions are satisfied, (f) how the null, weak and the strongenergy conditions are overcome, (g) how the generic perturbative corrections may be an-alyzed, (h) how the non-perturbative corrections may be analyzed, (i) how the non-localquantum terms may be understood, (j) how the four-dimensional Newton’s constant maybe kept constant , (k) how the positive cosmological constant may be generated by quan-tum corrections, (l) how the zero point energy gets renormalized in a non-supersymmetricbackground, (m) how the geometry and the topology of the internal compact space, whichis now a highly non-K¨ahler manifold, may be expressed (n) how the Bianchi identities aresatisfied in a time-varying background, (o) how the swampland criteria are averted, (p)how the early-time physics should be understood, (q) how the inflationary paradigm maybe recovered from our analysis, (r) how other related solutions like Kasner de Sitter ordipole-deformed de Sitter could be studied, etc.; all in a top-down (not bottom up!) stringtheory set-up.The situation is further complicated by the fact that one needs to solve almost all of the above problems to justify consistency of our background either as a supergravitysolution or as a Glauber-Sudarshan state. There doesn’t exist a simple solution that onlyanswers parts of the above set of questions, because then it will not lead to a well-definedsolution to the system. This all-or-none criterion makes the finding of de Sitter space instring theory a really hard problem. In our earlier works [21, 22, 23], we have managed to An interesting related question is whether the four-dimensional Newton’s constant gets renormalized .What is easy to infer is that the four-dimensional Newton’s constant remains time-independent, but it’srenormalization (or it’s running ) solely depends on the effective action whose perturbative and the non-perturbative parts at a given scale , as discussed in sections 3.1 and 3.2, will be elaborated in section 3.3.Somewhat intriguingly, as we shall discuss in section 3.3, the internal metric components do not receivecorrections to any orders in g as M bp for appropriate choice of the Glauber-Sudarshan state. – 5 –nswer most of the essential questions, so here we re-interpret all of them as a consequenceof being a Glauber-Sudarshan state. This re-interpretation turns out to be a completelydifferent beast in the sense that, as the reader herself or himself will find out, a completere-evaluation of the scenario is called for because a new set of rules needs to be laid outand a new set of computations needs to be performed. In our opinion, these have hithertonever been attempted, so the analysis will naturally get involved. These computations areessential to understand the full picture, or at least to appreciate the consistency of thefull framework, but the take-home message is surprisingly simple: viewing de Sitter as aGlauber-Sudarshan state and not as a vacuum, alleviates most of the problems associatedto entropy, TCC, stability etc.Let us concentrate on one such point from above, namely (d), i.e. how moduli stabi-lization may be understood when we expect the moduli themselves to vary with respectto time. Other points related to the non-perturbative corrections and the EOMs emanat-ing from the Schwinger-Dyson’s equations will be explained later as we progress in thetext. The remaining points, related to anomaly cancellations, flux quantizations etc., havealready been discussed in details in [21] so we will not dwell on them here.The reason for singling out (d) as opposed to the other points is because of its subtlety.Moduli stabilization requires a more careful analysis here because of the underlying Dine-Seiberg runaway problem [31]. Dine and Seiberg said that, once string vacua are left withunfixed moduli, they decompactify and quickly go to strong coupling. However, once moduliare fixed, vacua could be easily studied using perturbative string theory. However, note thatall of these discussions, as presented in [31], are for time-independent compactifications.Question is, how does this translate into the case when there are time dependences?In our case, as we mentioned in (d) above, once we fix the moduli at every instant of time, the Dine-Seiberg runaway can be stopped. The fixing of the moduli works inthe following way. Starting with the solitonic vacuum configuration, as in (2.1) with thecorresponding G-flux components to support it, the non-trivial quantum corrections cap-tured by H int , as described in section 2.4 onwards, the moduli get fixed without breakingsupersymmetry. This time-independent configuration with no running moduli forms oursupersymmetric vacuum configuration on which we study the fluctuations. The vacuumhowever is not a free vacuum: it’s an interacting vacuum from which we can construct ourGlauber-Sudarshan state (exactly how this is done will form the basis of sections 2.4 and2.5). Expectation values of the metric and the G-flux components will govern the behaviorof the moduli for the de Sitter case. We will call this the dynamical moduli stabilization .Interestingly, what we find in the time-independent case is that to order g s M p , i.e. tothe zeroth orders in g s (string coupling) and M p (Planck mass), although the Dine-Seibergrunaway is apparently stopped, there are still an infinite number of operators with nohierarchy when the internal degrees of freedom are time-independent. This clearly showus that there are no solutions to the EOMs and the vacua do not exist. Note that thishappens to any orders in g as M bp , and in particular for small g s , so is not a strong-couplingquestion at all !Thus the difficulty in generating a de Sitter solution lies not just on overcoming the– 6 –o-go and the swampland constraints, but also on the various technical challenges thatwe encounter along the path towards an actual realization of a background data either asexpectation values or as supergravity solutions. However as our earlier works [4, 21, 22, 23]and the present paper will justify, this is not an insurmountable problem. Solutions doexist and, with some efforts, may be determined precisely. Let us start with a simple example from quantum mechanics of a potential V ( x ) in 1 + 1dimensions that has at least one local minimum at x = a . The potential, near the vicinityof x = a , may be represented in the following standard way: V ( x ) = V ( a ) + 12 ω ( x − a ) + ∞ (cid:88) n =3 n ! λ n ( x − a ) n , (1.1)where λ n characterize the anharmonic terms. For λ n sufficiently small the low lying eigen-states near x = a will satisfy the consistency conditions: λ n (cid:104) ( x − a ) n (cid:105) << ω (cid:104) ( x − a ) (cid:105) , (1.2)with n = 3 , , .. , such that the wave-functions are that of a simple harmonic oscillator. Thedetails are rather well-known so we will avoid repeating them here, except to point outthat this simple analysis will form the basis of our construction in full IIB supergravity insection 2. The x = a point herein will form the solitonic vacuum for us.The simple analysis however hides a subtlety that is typically not visible from quantummechanics. Assuming the condition (1.2) to hold, i.e. we can ignore higher order terms,one would naively think that (1.1) is simple harmonic potential for a free theory. Thisis not correct: the simple harmonic oscillator term can in fact hide an infinite number ofinteractions with the soliton itself, as it becomes clear if we go to the field theory case inthe presence of an interacting Hamiltonian H int .We made a brief mention of the interacting Hamiltonian H int in section 1.1, but it is anabsolutely essential quantity to even construct the solitonic vacuum (2.1) and fluctuationsover it. All of these will be elaborated as we go along, but here we would like to constructa toy example in quantum field theory that captures some of the salient features of theactual construction in a simpler fashion.The basics of the toy example resides in the soliton physics, and there are many excel-lent textbook treatments on the subject − for example [32] − but here we will generalizethis a bit to capture the influence of the interacting Hamiltonian. To avoid conflicting withthe Derrick’s theorem [33], we will concentrate only in 1 + 1 dimensions with a single scalarfield ϕ ( x ) where x = ( x , t ). Let us assume that the potential for the scalar field is: V ( ϕ ) = (cid:88) n,m C nm { ∂ n } ϕ m ≡ (cid:88) n,k,m C ( k ) nm ϕ n (cid:16) ∂ k ϕ (cid:17) m (1.3)= (cid:88) m C m ϕ m + (cid:88) n,m C (1) nm ϕ n ( ∂ϕ ) m + (cid:88) n,m C (2) nm ϕ n (cid:0) ∂ ϕ (cid:1) m + (cid:88) n,m C (3) nm ϕ n (cid:0) ∂ ϕ (cid:1) m + ..., – 7 –hich is basically a very simplified version of (3.2) that we shall encounter in section 3.1.In constructing (1.3) we have ignored both time derivatives and supersymmetry as neitherare very essential to understand the dynamics here. The derivative terms are all suppressedby the coupling constants C ( k ) nm which, in the language of (3.2), are proportional to g | a | s M bp .Here of course we will simply assume that C ( k ) nm << g s or M p .We will also assume that there is a solitonic vacuum given by ϕ = ϕ ( x ).Existence of the solitonic vacuum implies that we are looking at the minima of thetotal potential, which is the combination of the potential (1.3) and the contribution fromthe kinetic term. The fluctuation over the solitonic vacuum can be represented as η ( x , t )which is a function of both space and time. In the presence of the fluctuation, we canexpress the field as: ϕ ( x , t ) = ϕ ( x ) + η ( x , t ) ≡ ϕ ( x ) + (cid:90) d k f k ( t ) ψ k ( x ) ≈ ϕ ( x ) + (cid:88) k f k ( t ) ψ k ( x ) , (1.4)where in the last equality we have assumed discrete momenta, and f k ( t ) is generic amplitudenot necessarily on-shell. Note that the integral is over d k and not over d k = d k dk , whichis of course what we expect. Question is whether we can determine the function ψ k ( x ). Itturns out that the function ψ k ( x ) satisfies the following Schr¨odinger equation: (cid:34) −∇ + (cid:88) m C mm C ϕ m − + (cid:88) n,m,p C ( p ) nm ϕ n ( ∂ p ϕ ) m (cid:18) n C ϕ + nm∂ p ϕ ∂ p ϕ − m C ( ∂ p ) ( ∂ n ϕ ) (cid:19)(cid:35) ψ k ( x ) = ω k ψ k ( x ) , (1.5) with eigenvalue ω k and p ≥
1. We have used a flat metric, and the powers of thederivatives, i.e ∂ p , are only along x , so there should be no confusion. A missing factor of 2in the first term of (1.5), to allow − ∇ so that (1.5) may indeed look like a Schr¨odingerequation, can be easily inserted in by a simple redefinition of the x coordinate. This is astandard manipulation (see [32]), so we will not worry about it and call (1.5) simply as theSchr¨odinger equation with a highly non-trivial potential.The potential that enters the Schr¨odinger equation is not the potential in (1.3), al-though it is related to it, but the crucial question is how do the eigenstates ψ k behavein this potential. Before we discuss this, we should point out that, plugging (1.5) in the1 + 1 dimensional field theory action immediately reproduces the standard simple harmonicoscillator action in the following way: S = 12 (cid:90) dtd k (cid:32)(cid:12)(cid:12)(cid:12) ˙ f k ( t ) (cid:12)(cid:12)(cid:12) − ω k (cid:12)(cid:12)(cid:12) f k ( t ) (cid:12)(cid:12)(cid:12) − V ( ϕ ) + .... (cid:33) , (1.6)where the dotted terms are O (cid:0) | f k | (cid:1) interactions. In this form the potential part of theaction matches precisely with the quantum mechanical result from (1.1), but the difference Note that the M-theory background (2.1), or it’s IIB counterpart, is a soliton in only a loose sense. Itis of course clear that there does exists a vacuum configuration of the form R , × T with no G-flux, so(2.1), with it’s corresponding G-flux, forms the nearby vacuum configuration with non-zero energy. – 8 –hould be clear: (1.6) is derived from an highly interacting theory whereas (1.1) is a simplequantum mechanical result. Despite that we will call the vacuum from (1.6) as the free vacuum (or sometime the harmonic vacuum) to distinguish it from the interacting vacuumto be discussed later.The eigenstates ψ k ( x ) and the eigenvalues ω k are important because they will decidethe subsequent behavior of the 1 + 1 dimensional field theory. Typically eigenstates of aSchr¨odinger equation are divided into three categories: (a) zero modes with ω k = 0, (b)discrete levels with ω k given by a set of discrete integers, and (c) continuum levels where ω k is related to k by an on shell condition. The latter is what we want, but we need toworry about the zero modes and the discrete states. What do they mean here?In standard solitonic theory, the zero modes are related to the motion of the solitonitself, and they are typically used to quantize the soliton. For our case, the soliton will berelated to the vacuum metric configuration (2.1) and the zero modes should appear as thetranslation or the rotational modes of the internal metric that do not change the energy ofthe system . Do they exist? For the simple scalar field theory case one can easily show thatif the solitonic solution ϕ ( x ) belongs to an equipotential curve consisting of the family ofmutually translated solitons ϕ ( x − a ), then there does appear one zero mode ψ ( x ) thattakes the following form: ψ ( x ) = ∂ϕ ( x ) ∂ x , (1.7)which should solve the equation (1.5) with ω k = 0. For the case we want to concentrateon, i.e the metric (2.1) with the corresponding G-fluxes, one needs to check whether suchzero modes can appear. There are also the K¨ahler and the complex structure moduli ofthe internal metric which tell us that we can change the metric without costing any energyto the system. These moduli are governed by a Lichnerowicz type of equation (now inthe presence of metric, fluxes and quantum corrections). Fluctuations over these metric(and flux) configurations would now have their corresponding Schr¨odinger equations like(1.5). All these class of equations should be related to each other and therefore one couldassociate a combination of the zero modes of these Schr¨odinger equations to these modulithemselves. Clearly this will make the system much more complicated, so to avoid this aswell as the Dine-Seiberg runaway [31], we want the moduli to be fixed. The moduli arefixed in the presence of quantum terms, and therefore we expect those zero modes thatcorrespond to the K¨ahler and the complex structure moduli to not arise in the presenceof sufficient number of quantum terms in (1.5). For a generic potential like the one thatappears in (1.5) this may be hard to show, but there are alternative ways (see for example[34, 35]) to justify this (more on this a bit later).The discrete modes correspond to the bound state of the mesons, which are basicallythe fluctuations of the scalar field, with the soliton. These mesons get trapped inside thepotential of the soliton and the discrete states show that the probability amplitudes peaknear the centre of the soliton. For the background (2.1), again there is no such simpleinterpretation because we will treat the fluctuations separately and not consider themas getting bound by the soliton. Of course these differences arise because the classical– 9 –ackground (2.1) and the corresponding G-flux shares many similarities with a soliton, butis a soliton only in a loose sense (see footnote 2).All our above discussion points out that it is the continuum level of the Schr¨odingerequation (1.5) that concerns us here, although we will interpret these wave-functions asthe fluctuation wave-functions and not scattering states with the soliton. Again the slightdifference in interpretation stems from our loose identification of (2.1) to an actual soliton.The set of the continuum wave-functions ψ k ( x ) takes the following form: ψ k ( x ) ≡ exp ( i kx ) G k ( x )lim x →±∞ ψ k ( x ) = exp [ i kx ± iσ ( k )] , (1.8)where G k ( x ) is a non-trivial complex function of ( k , x ) which for large x approximates toa phase σ ( k ). In this sense the continuum level resembles somewhat the scattering states.The analysis for the actual case with (2.1) will be much more involved, but fortunately, aswe shall see, we will not have to determine the functional forms for ψ k ( x ) explicitly. The paper is organized in the following way. Broadly, section 2 studies the backgroundfrom solitonic point of view, i.e. from the M-theory uplift of the supersymmetric warpedMinkowski background in (2.1); whereas sections 3.1 and 3.2 studies the same directly using(2.4), which is the M-theory uplift of the IIB de Sitter background of (2.2). In section 3.3 wederive the de Sitter results in two ways, one, by taking expectation values over the Glauber-Sudarshan state and two, by solving the Schwinger-Dyson’s equations. Thus section 3.3serves as a culmination and synthesis of the results accumulated from sections 2, 3.1 and3.2. Section 4 serves as a vantage point to analyze many of the important properties of deSitter space from the point of view of the Glauber-Sudarshan state, namely trans-Planckiancensorship, quantum swampland and de Sitter entropy.Let us now go to a more detailed survey of various sections of the paper. In sec-tion 2.1 we lay out the formalism of the Glauber-Sudarshan state and discuss how oneshould construct it using the momentum modes of the theory. In particular we discuss theprecise wave-function of the Glauber-Sudarshan state using configuration space variablesassociated with the metric components of the solitonic vacuum. Other properties, like theSchr¨odinger wave-functions, oscillatory behavior, and the study of the zero modes are alldiscussed here.Section 2.2 studies the fluctuations over a de Sitter background directly from our soli-tonic configuration. Since the de Sitter space itself is a state over the solitonic background,the fluctuations should also appear from a corresponding state. In this section we discusshow such state should be constructed by combining the various oscillatory states over thesolitonic vacuum. We also discuss how the fluctuations over a de Sitter vacuum could bereinterpreted as an artifact of Fourier transform over the solitonic vacuum. This meansthe time-dependent frequencies that we see from the fluctuations over a de Sitter vacuum is a consequence of the linear combinations of the modes over the solitonic vacuum. Thisprovides not only an answer to the trans-Planckian issue because the frequencies over the– 10 –olitonic vacuum are time-independent, but also provides a way to tackle the Wilsonianintegration.In section 2.3 we identify the state that defines the fluctuations over de Sitter space − viewed as a Glauber-Sudarshan state − as the Agarwal-Tara [36] state. The Agarwal-Tarastate, sometime also called as the Agarwal state, is constructed by adding gravitons to theGlauber-Sudarshan state and is therefore popularly known as the graviton added coherentstate (GACS). We construct a generic operator, controlled by a coupling parameter, thatwhen acting on the Glauber-Sudarshan state creates the necessary Agarwal-Tara state.So far we have used the free (or the harmonic ) vacuum to construct the Glauber-Sudarshan state. This cannot quite be the full picture because there is no free vacuum inan interacting theory like M-theory. Thus we have to construct our Glauber-Sudarshanfrom an interacting vacuum | Ω (cid:105) . Such a construction is laid out in section 2.4, we we startby first shifting the interacting vacuum using a displacement operator. The constructionof the displacement operator is in itself a non-trivial exercise because of the interactionsthat mix all the momentum modes of the metric and the flux components. Expectationvalues of the metric operators over such shifted interacting vacuum, which we call as the generalized Glauber-Sudarshan state, are carried out using the path integral formalism.This is elaborated in section 2.5. The path integral formalism is particularly useful inanalyzing the expectation values and we show that we can reproduce the expected metricconfiguration in (2.4) from there, up to O (cid:18) g | a | s M bp (cid:19) corrections. These corrections are sub-leading, and their presence is because of our choice of the generic form of the displacementoperator. There does exist a specific choice of the displacement operator that reproduces(2.4) as expectation value precisely, without any extra corrections. To develop that requiresmore preparation, and we postpone it till section 3.3.The path integral formalism is also powerful to study the expectation values of themetric components over the Agarwal-Tara state, viewed as an operator acting on the gen-eralized Glauber-Sudarshan state. The answer that we get clearly shows not only thatwe can reproduce the fluctuation spectrum over a de Sitter space, but also the fact thatthe fluctuation spectrum is indeed an artifact of the Fourier transform over our solitonicvacuum.What we haven’t tackled so far is the contributions from the G-flux components. Thisis discussed in section 2.6. Although these contributions make the system a bit morecomplicated, their presence is necessary for the self-consistency of the Glauber-Sudarshanstate. For example they help us to understand how supersymmetry is broken spontaneouslyby the state while the solitonic vacuum remains perfectly supersymmetric. The new modesthat are created by fluctuations over the G-flux background in the solitonic vacuum nowmix non-trivially with the modes from the metric components. This leads to complicatedinteractions and therefore the stability of the Glauber-Sudarshan state becomes an issue.These interactions are the main focus of sections 3.1 and 3.2, and the question of thestability of the Glauber-Sudarshan state is finally resolved in section 3.3. To study theinteractions we change the gear a bit by using the background (2.4) directly instead ofgoing via the configuration (2.1). There is a definite advantage of using such a procedure– 11 –s will become clear from the computations in sections 3.1 and 3.2.The quantum corrections are a bit non-trivial to deal with because there are an infinite number of possible local and non-local − that include the perturbative, non-perturbativeand topological − interactions. In section 3.1 we classify the perturbative contributionsusing the formalism developed earlier in [21, 22]. The non-perturbative corrections havenot been discussed before, and we detail them in section 3.2. These corrections comegenerically come from the branes and the instantons, and we show that certain aspects ofthese corrections may be extracted from the non-local interactions discussed in [21, 22].The non-perturbative contributions from the instantons come from both M2 and M5-instantons, and we show that it’s only the M5-instantons contribute here. The M5-instantons’ contributions are further classified by the BBS [37] and KKLT [5] type in-stantons. Their contributions are discussed in sections 3.2.1 and 3.2.3 respectively. Thecontributions from the seven-branes and in particular the fermionic terms on the seven-branes are discussed in details in section 3.2.2.All these perturbative and non-perturbative quantum corrections contribute to theequations of motion (EOMs). In section 3.3 we show that these EOMs appear from aclass of Schwinger-Dyson’s equations (SDEs) [38]. Interestingly, the SDEs for our casesplit into two sets of equations. One set of equations are completely expressed in termsof expectation values over the solitonic vacuum. Since, as we discussed in sections 2.4and 2.5, the expectation values of the metric and the G-flux components over the solitonicconfiguration (2.1) reproduces the background (2.4), along with it’s G-flux componentsfrom section 2.6, these SDEs give rise to the M-theory EOMs for the background (2.4)!This immediately implies that all the computations that we did in sections 3.1 and 3.2appear now as quantum contributions to the SDEs.There is also a second set of SDEs that relate the Faddeev-Popov ghosts, the displace-ment operator and the expectation values of the variations of the total action with respectto the field variables. These SDEs are in general hard to solve and we leave them for futureworks. In section 3.4 we give a brief discussion on supersymmetry breaking and other re-lated effects. One of the important question that we tackle in this section is the connectionbetween the supersymmetry breaking scale and the cosmological constant Λ. We showthat in general there is a large hierarchy between them, implying that the supersymmetrybreaking scale could be large yet Λ could remain relatively small. Of course the exact valueof Λ would depend on the values of the fluxes and the quantum terms, which would onlybe determined if we solve all the SDEs exactly, but what we argue here is that the preciseform, which may be extracted from our earlier work [21], appears to have no contributionsfrom the ground state energies of the bosonic and the fermionic degrees of freedom. Wealso elaborate on the moduli stabilization and discuss what happens when we go to thestrong coupling limit of type IIB.Once we construct our de Sitter solution using the Glauber-Sudarshan state, we ex-amine some of its properties, especially with respect to the swampland. The aim of theswampland has been to argue against the existence, and especially the stability, of de Sittersolutions from general quantum gravity arguments that having nothing to do with stringtheory constructions in particular. In our work, we first establish how our solution manages– 12 –o escape the so-called (refined) de Sitter swampland condition and, more importantly, theTCC. We show that the time for which our solutions remain well-defined is compatiblewith the time-limit set by the TCC. We do not argue for or against the criterion set by theTCC but rather find that, quite remarkably, our Glauber-Sudarshan state naturally satis-fies it. However, more pertinently for us, we shall show why the trans-Planckian problemdoes not even apply to our coherent state description. This is because our setup is alreadyincorporates a UV-complete theory – string theory – and have an underlying Minkowskispace-time. In section 4.1, we shall elaborate on how our solution escapes the TCC basedon these two conditions. More general field-theoretic obstructions against the stability ofde Sitter space-times essentially arise from the arbitrariness of the choice of the vacuumfor fluctuations on top of de Sitter. It has long been extensively debated what is the right choice for this vacuum. While some have argued in favour of the Bunch-Davies vacuum,other have advocated for, say, de Sitter-invariant vacuum like the α -vacuum [39, 40]. Weshall demonstrate in section 4.2 is that the main reason for the problems emerges for thesevacua is due to the complicated time-dependence of the mode functions corresponding tothem. And thankfully, this is exactly what is not the case for our solution since even fluc-tuations over de Sitter is constructed over the interacting vacuum in Minkowski spacetimeand, therefore, the time-dependences arise as an artifact of Fourier transforms over theGlauber-Sudarshan state.Finally, in section 4.3, we interpret the usual Gibbons-Hawking entropy for de Sitter asan entanglement entropy between the modes, on top of the warped-Minkowski background,which construct the Glauber-Sudarshan state itself. We then show how this entanglemententropy remains for our de Sitter solution. This is mainly due to two factors – (a) havingthe de Sitter symmetries being emergent in our scenario which remain valid for a finitetime-period, and (b) there is necessarily an interaction Hamiltonian H int for us. If we takethe limit H int →
0, we find that the entanglement entropy goes to infinity and, on theother hand, for the same limit, our Glauber-Sudarshan state cannot be constructed andwe get back flat space-time. In this sense, we find that there is a nice consistency for whywe manage to find a finite entropy for de Sitter, while laying down the path for gettingcorrections beyond the semiclassical result.
Throughout the paper we have used the mostly-plus convention although in a couple offield theory computations we have used the mostly-minus convention so as to comply withknown results. They will be indicated as we go along. Similarly, the Hubble parameterwill be denoted by H whereas the warp-factor will be denoted by H = H ( y ). The Hubbleparameter features prominently in section 4, so this should not be a cause of confusion(in any case it’ll be properly indicated which is which). The eleven-dimensional Planckconstant will be denoted by M p whereas we will use M Pl to denote four-dimensional Planckconstant.More importantly, in section 2 we will mostly use the solitonic background (2.1), andtherefore the fields will be denoted by ( g MN , C MNP ), whereas the operators will be denotedby bold faced letters, i.e. ( g MN , C MNP ), unless mentioned otherwise.– 13 –n sections 3.1 and 3.2 we will use the background (2.4) and therefore they will involve g s dependent quantities. Both these sections only involve fields and to distinguish themfrom the field variables of the solitonic background, we use bold faced letters to write them,i.e ( g MN , G MNPQ ) will denote fields associated with the uplifted de Sitter background (2.4)in M-theory.In sections 3.3 and 3.4 we have to use both fields and operators of the solitonic back-ground as well as the fields of the uplifted de sitter background (2.4). Our convention forthis section then is the following: all fields and operators over the solitonic vacuum areexpressed using un-bolded letters, for example g MN denotes a field and (cid:104) g MN (cid:105) σ denotes theexpectation value of an operator over some state | σ (cid:105) constructed over the solitonic vacuum.Again which is which should be clear from the context. The bold faced letters are reservedfor the fields associated with the uplifted de Sitter background (2.4) in M-theory as wehad in sections 3.1 and 3.2. This can be made clear by an example: (cid:104) g MN (cid:105) α = g MN , whichimplies that the expectation value of the metric operator over the Glauber-Sudarshan state | α (cid:105) gives the warped metric component of the de Sitter space, namely (2.4).
2. de Sitter space as a Glauber-Sudarshan state
Our analysis of the toy example in section 1.2 for a 1 + 1 dimensional field theory providedthe necessary groundwork on which we can build our theory. Our aim would to allow fora solitonic configuration in IIB, which would be stable supersymmetric solution. Due tocertain technical efficiency, as discussed below, it is better to uplift the configuration toM-theory. This is as discussed in [21], wherein we will realize the solitonic vacuum fromM-theory instead of type IIB, as: ds = 1 (cid:112) h ( y, x i ) (cid:0) − dt + dx + dx (cid:1) + (cid:112) h ( y ) g (0) MN dy M dy N , (2.1)where h ( y ) and h ( y, x i ), i = 1 ,
2, are the warp-factors and g (0) MN ( y ) is the metric of theinternal eight-manifold. Although the choice of M-theory over type IIB is mainly becauseof the compactness of the representations of the degrees of freedom (the total degrees offreedom remains the same on either sides), the fact that M-theory allows a well-defined lowenergy effective action whereas there is no simple action formalism for the IIB side, providesa better motivation to dwell on the M-theory side. Additionally, since our analysis will relyheavily on the path integral formalism, that in turn relies on the presence of an effectiveaction, the M-theory uplift is more useful here. Minor compromise, like compactifying the x direction, will not have any effect on the late time physics that we want to study here.Coming back to our simple harmonic oscillator from section 1.2, we see that, in additionto the solitonic solution x = a , there are other solutions of the form x = a + A cos ωt ,for arbitrary choices of the constant A . These are time-dependent solutions having anoscillatory part, but they solve the EOMs. On a solitonic vacuum x = a , the oscillatorypart may be realized as a coherent state [28]. In QFT, such configurations are the Glauber-Sudarshan states [26, 27] and their presence are augmented by the fact that they solve the– 14 –OMs. In fact it is the combination x = a + A cos ωt that solves the EOMs, so we can tryto realize the type IIB de Sitter solution of the form: ds = 1Λ( t ) √ h ( − dt + dx + dx + dx ) + √ h (cid:16) F ( t ) g αβ ( y ) dy α dy β + F ( t ) g mn ( y ) dy m dy n (cid:17) , (2.2) as a Glauber-Sudarshan state over a Minkowski background. Note that the four-dimensionalpart of (2.2) is a de Sitter space with a flat-slicing when Λ( t ) = Λ | t | and therefore thetemporal coordinate covers −∞ ≤ t ≤
0, implying t → h i in (2.1) to be related. The relation is rathersimple: h ( y ) = h ( y ), but here we generalize this somewhat by keeping h ≡ h ( y, x i ) and h ≡ h ( y ) unequal. The equality is a special case dealt in great details in [41] and [34].Secondly, the solitonic background (2.1), with a compact internal eight-dimensionalspace M , can only be supported in the presence of G-fluxes. If we denote ( y m , y α )respectively to be the coordinates of the six-dimensional base, with ( α, β ) = (4 ,
5) and( m, n ) = (6 , , ,
9) as M × M of M such that: M ≡ M × (cid:18) M × T G (cid:19) , (2.3)with ( y a , y b ) denoting the coordinates of the fiber torus ( G is the isometry group of thetorus), then we need G-fluxes with components on both the base and the fiber, as wellas components like G ijm and G ijα , all functions of the six-dimensional base coordinates[41, 34, 21].All these imply that the fluctuations over the background (2.1) as well as over the G-flux components that satisfy the EOMs allow more non-trivial time-independent Schr¨odingertype equations whose solutions provide the fluctuation modes of the spectra, at least if wewant to bring them in the simple harmonic form satisfying the condition (1.2) . The identi-fication to (1.1), or even to (1.2), is more subtle as fluctuations along both space-time as wellas the internal directions need to be accounted for, implying that the Glauber-Sudarshanstates are not as simple as they were for the case with electromagnetic fields [26, 27]. Nev-ertheless, once we know the corresponding Schr¨odinger wave-functions we can at least tryto use them to determine the Fourier modes of the metric and the G-flux components. Theadditional leverage that we get here is from the existence of the coherent states themselves(which we will justify a bit later). Assuming that the coherent states may be constructed As it happens in any quantum field theory, the fluctuations never satisfy the full equations of motion.They only satisfy linear equations appearing from the quadratic parts of the action when expressed interms of the fluctuating fields. As an example for a scalar field fluctuation δϕ , written as δϕ ( x, y, z ) = (cid:82) d k f k ( t ) ψ k ( x , y, z ), generically it’s only the ψ k ( x , y, z ) piece that becomes non-trivial over the solitonicbackground (2.1). See section 1.2 for more details. – 15 –or all modes, at least in the case where we can bring the individual mode-dynamics tothe simple-harmonic case, this will justify that in the configuration spaces of each modesthere are simple oscillatory motion (at least for both real and the complex parts). This isthen obviously consistent with the time-dependent parts of the corresponding Schr¨odingerwave-functions which become ψ k ( x ) , η k ( y, t ) , ξ k ( y, t ) and ζ k ( z, t ) respectively for the 2 + 1dimensional space-time, the internal six-manifold M × M and the fiber torus for everymode k and at any given time t . There is one issue that we kept under the rug so far, and has to do with the F i ( t ) factorsin (2.2). These factors are essential if we want to realize the IIB configuration (2.2) as aGlauber-Sudarshan state, simply because coherent states generically cannot produce time-independent configurations! Thus what we want in M-theory is a configuration of thefollowing form : ds = 1 (cid:16) Λ | t | √ h (cid:17) / ( − dt + dx + dx ) + e B ( y,t ) g αβ dy α dy β + e B ( y,t ) g mn dy m dy n + e C ( y,t ) g ab dx a dx b , (2.4) that encapsulates all the essential features of the IIB background (2.2) with appropriatechoices of the coefficients ( B i ( y, t ) , C ( y, t )). The way we have expressed (2.4) suggestsgeneralities beyond (2.2), although one expects: C ( y, t ) ≡
12 log (cid:16) [Λ( t )] / [ h ( y )] / (cid:17) , (2.5)if one wants to preserve the full de-Sitter isometries in 3 + 1 dimensional space-time in theIIB side. With this in mind, our first guess for the four set of Fourier components are: (cid:101) g µν ( k ) = (cid:90) d x (cid:16) Λ | t | √ h (cid:17) / − h / h − ψ ∗ k ( x ) η µν (cid:101) g αβ ( k ) = (cid:90) d ydt (cid:113) g (0 , (cid:16) e B ( y,t ) g αβ − h / g (0) αβ (cid:12)(cid:12)(cid:12) base (cid:17) h − / η ∗ k ( y, t ) (cid:101) g mn ( k ) = (cid:90) d ydt (cid:113) g (0 , (cid:16) e B ( y,t ) g mn − h / g (0) mn (cid:12)(cid:12)(cid:12) base (cid:17) h − / ξ ∗ k ( y, t ) (cid:101) g ab ( k ) = (cid:90) d zdt (cid:113) g (0)fibre (cid:18) h / Λ / | t | / δ ab − h / g (0) ab (cid:12)(cid:12)(cid:12)(cid:12) fibre (cid:19) h − / ζ ∗ k ( z, t ) , (2.6) A small puzzle appears now that is worth mentioning at this stage. In M-theory, the internal eight-manifold is always time-dependent if the four-dimensional space in the type IIB side has de Sitter isometries.As such a coherent state construction in M-theory should work whether or not the internal six-dimensionalspace in IIB is time-dependent. Why do we then need the internal space metric in IIB to take the form(2.2)? The resolution of the puzzle will require a more detailed understanding of the existence of a coherentstate in M-theory, that we shall indulge in later (see section 3.3). For the time-being we will assume boththe internal six and the eight manifolds in IIB and M-theory respectively to be time-dependent. – 16 –ith g (0 ,q )base denoting the classical base metric of the six-manifold expressed as M × M . Inwriting (2.6) we have ignored many subtleties that we should clarify. First, the reason fortaking Fourier transforms is because the construction of Glauber-Sudarshan state is mosteasily expressed in terms of the Fourier components, as we shall see soon. Second, because ofthe solitonic pieces, (2.6) is partly off-shell. Third, the modes ( ψ k ( x ) , η k ( y, t ) , ξ k ( y, t ) , ζ k ( z, t ))aren’t necessarily as simple as we presented here. If we write x ≡ ( x , t ), i.e. separate thespatial and temporal coordinates, then the four modes that actually appear from the un-derlying Schr¨odinger type equation are typically of the form :Ψ k ( x , y, z, t ) ≡ ψ k ( x , y, z, t ) , η k ( x , y, z, t ) , ξ k ( x , y, z, t ) , ζ k ( x , y, z, t ) , (2.7)where ψ k ( x , y, z, t ) denotes the set of modes governing the dynamics in 2 + 1 dimensionalspace-time. Similarly ξ k ( x , y, z, t ) denotes the set of modes governing the dynamics on M internal space, and so on. These subtleties will only become relevant in section 3.3,so for the time being we will avoid over-complicating the system by assuming only singleset of modes representing the directions respectively. Note that we have isolated the t dependence on each modes and y ≡ ( y α , y m ). The reason is that the t dependences ofeach modes should be of the form exp (cid:16) iω ( a ) k t (cid:17) with a ≡ ( ψ, η, ξ, ζ ) signifying the differentmodes. Such a temporal dependence is absolutely essential if the system has to allow for acoherent state description. Alternatively, this boils down to the familiar decomposition ofthe metric fluctuations as: g µν ( x, y, z ) = η µν h / ( y, x ) + (cid:90) d k (cid:101) g µν ( k , t ) ψ k ( x , y, z ) , (2.8)with similar decompositions for the other internal components. Note two things: one, theintegral is over d k and not over d k , and two, the appearance of a generic (cid:101) g µν ( k , t ) andnot just (cid:101) g µν ( k ) from (2.6). They are of course expected consequences of any standard fieldtheory so we will refrain from elaborating further on them. Additionally, fixing the valuesof ( µ, ν ) would lead to three set of fields, with each field having an infinite set of modes withan infinite number of harmonic oscillator states for each modes. All these follow standardresults and if we, with some abuse of notations, define ψ k ( x, y, z ) = ψ k ( x , y, z )exp (cid:16) iω ( ψ ) k t (cid:17) ,then the “Fourier” modes in the first line of (2.6) follow naturally once we fix y = y and z = z to some slice in the internal space. Such a choice of slice simplifies the underlyinganalysis but has no physical consequence. Thus we could easily replace ψ k ( x ) by ψ k ( x, y, z ) We have used a simplifying normalization condition for the modes in (2.6). The correct on-shell nor-malization condition for all the modes in (2.7) should be: (cid:90) d x Ψ k ( x , y, z, t )Ψ ∗ k (cid:48) ( x , y, z, t ) h − ( x , y ) h / ( y ) ≡ δ ( k − k (cid:48) ) δ ( ω ( a ) k − ω ( a ) k (cid:48) )However the compactness of the internal eight-manifold will simplify this and one can bring it in the formused for (2.6) if one further restricts to a slice in the internal space. As mentioned later, such restriction isnot essential. Thus naturally identifying k in (2.6) (and also in (2.9)) as k ≡ ( k , ω k ). – 17 –tc in (2.6) giving us the following on-shell pieces (we will deal with the off-shell part soon): (cid:101) g µν ( k ) = (cid:90) d x (cid:113) g (0)base (cid:16) Λ | t | √ h (cid:17) / h − ( x , y ) h / ( y ) ψ ∗ k ( x, y, z ) η µν (2.9) (cid:101) g αβ ( k ) = (cid:90) d x (cid:113) g (0)base (cid:16) e B ( y,t ) g αβ (cid:17) h − ( x , y ) h / ( y ) η ∗ k ( x, y, z ) (cid:101) g mn ( k ) = (cid:90) d x (cid:113) g (0)base (cid:16) e B ( y,t ) g mn (cid:17) h − ( x , y ) h / ( y ) ξ ∗ k ( x, y, z ) (cid:101) g ab ( k ) = (cid:90) d x (cid:113) g (0)fibre (cid:16) h / Λ / | t | / δ ab (cid:17) h − ( x , y ) h / ( y ) ζ ∗ k ( x, y, z ) , where the determinant g (0)base ≡ g (0 , g (0 , g (0)fibre as defined for (2.6). The slight differencesfrom (2.6) are significant because they would eventually determine the Fourier components.The above discussion tells us that the modes in the theory are typically the time inde-pendent modes for any k , and the time-dependences appear from the harmonic oscillatorstates that have energies in odd and even multiples of ω ( a ) k . However there are other sub-tleties that appear here that need some elaborations. First, let us concentrate on the zeromodes, i.e. the possibility of modes satisfying: ω ( ψ ) k = ω ( η ) k = ω ( ξ ) k = ω ( ζ ) k = 0 . (2.10)These are the troublesome modes in the theory that would lead to the Dine-Seiberg [31]runaway problem, even at the level of the solitonic vacuum (2.1). These zero modes appearfrom the complex and the K¨ahler structure moduli and therefore they have to be fixed tomake sense of the underlying theory. In the standard solitonic theory, these zero modeshelp us to quantize the soliton itself, but here we will have to deal with them differently.In fact the time-independent G-fluxes that we added in the theory would help us achieveour goal by creating the necessary superpotential [34, 35, 42] .Such an approach now ties two things together. One, the fact that the ( ψ k , η k , ξ k , ζ k )modes in the solitonic background are not of the kind exp ( ± i k · x ) , exp ( ± i k · y ) and As elaborated in section 1.2, this identification of the K¨ahler and the complex structure moduli to thezero modes (2.10) is more subtle. To illustrate the point, let us consider the wave-function ξ k ( x , y, z ) whosezero-modes may be denoted by the set ξ (0) k ( l ) with l denoting the parameter associated with the moduli(for example in the Calabi-Yau case l will parametrize ( h , h ) moduli). The set of zero modes satisfySchr¨odinger equations of the form: (cid:2) −∇ + V ( l ) ( x , y, z ) (cid:3) ξ (0) k ( l ) = 0 , which is similar to the Schr¨odinger equation (1.5) with two main differences: one, the potential V ( l ) ( x , y, z )is more involved than what appears in (1.5), and two, there are not one set of Schr¨odinger equations, but l set of them. This representation is not unique, and in fact there are an infinite possible such choices,depending on what values of the moduli we choose (i.e what values of the K¨ahler and the complex structuremoduli we choose). Clearly a linear combination of ξ (0) k ( l ) will contain the information of these moduli,although extracting them might be harder. However if the moduli are stabilized then the only zero modesthat we need to worry about are the translation and the rotations of the background (2.1) with it’s G-fluxcomponents. – 18 –xp ( ± i k · z ); and two, the presence of the G-fluxes. The latter leads to − and the formeris a consequence of − interactions . Thus the underlying theory that we discuss here cannotbe a free theory! This also means that the coherent states that we study here are not thecoherent states of a free theory.This is where we differ from the standard coherent state constructions in QuantumField Theories, but now the pertinent question is the source of the interactions themselves.Where are the interactions coming from? This is already answered in some sense in [41, 34]and more recently in [8]. The answer lies in the compactness of the internal manifold, bothin the IIB and in the M-theory sides. From M-theory, it is easy to argue that a genericcompact internal space cannot be supported in the absence of the G-fluxes [41, 43, 44, 34].Once G-fluxes are switched on, they automatically allow higher curvature topological termslike C ∧ X , where X involve fourth order curvature terms [41]. To the same order, non-topological terms, like: S ntop = M p (cid:90) d x √ g (cid:18) t t − (cid:15) (cid:15) (cid:19) R + O (cid:104) ( ∂ G ) (cid:105) + ...., (2.11)are switched on simultaneously, where the dotted terms are the mixed terms. Similar storyunfolds in the type IIB side also as emphasized recently in [8]. The outcome of our discus-sion then is the following. There is no simple free field theory description in the presence ofG-fluxes and/or compact internal manifolds. The latter is absolutely necessary to allow fora finite Newton’s constant in the non-compact directions. As a further consequence of ourstatement above, the modes ( ψ k , η k , ξ k , ζ k ) then become highly non-trivial but thankfullyremain time-independent .The underlying Schr¨odinger equation that appears from the interacting Lagrangianin the M-theory side, could also allow bound state solutions, in addition to the zero modespectra (2.10). These, if present, should be interpreted as the discrete excited states of thefull solitonic backgrounds (2.1), with various choices of the internal manifolds, themselves.We will not worry about them for the time being, and concentrate only on the continuum oflevels parametrized by the eigenfunctions ( ψ k , η k , ξ k , ζ k ). These are essentially the modesthat appear, once we juxtapose them with the temporal behavior exp (cid:16) iω ( a ) k t (cid:17) , in theFourier transforms (2.9). Once we go to the Euclidean picture, where k → k E , we canintegrate out the modes lying between M ≤ k E ≤ Λ UV to write the Wilsonian action at agiven scale M and with a UV cut-off Λ UV . This scale M could in principle be identified withM p , and in that case the effective action at that scale would match-up with the effectiveaction that we elaborated in great details in [21, 22] (see [23] for a review on this).More subtlety ensues, mostly associated with the oscillatory behavior of the coherentstates that allowed us to interpret the Fourier modes in (2.6) as ( ψ k , η k , ξ k , ζ k ) in the As pointed out in section 1.2, the harmonic oscillator regime will henceforth be termed as the free vacuum | (cid:105) here, and | Ω (cid:105) , which we will deal in section 2.4, will be the interacting vacuum with the fullanharmonicity brought in, unless mentioned otherwise. Much like the excited states of an electron in a hydrogen atom. The bound state spectra of theelectron precisely spell out these states. The continuum levels of the electron, on the other hand, have aninterpretation of scattering states. See also section 1.2. – 19 –rst place. For example, the way we have expressed (2.6), once added to the solitonicbackground (2.1), it appears to simply replace the solitonic background (2.1) by the time-dependent background (2.4), which is the uplift of the IIB de Sitter background (2.2). Howis this then any different from simply taking (2.2) as the non-supersymmetric vacuum intype IIB? The answer lies in the distinction between classical solution and quantum states.The M-theory background (2.4), or it’s IIB counter-part (2.2), does not appear just as aclassical solution here. Rather the probablity amplitude in a given coherent state for any given mode derived from (2.6) peaks at a specific value which once added to the classicalsolitonic solution (2.1) reproduces the background (2.4), or equivalently (2.2) in IIB. Thisreinforces the main point of our paper, namely, (2.4) in M-theory, or equivalently (2.2)in type IIB, cannot appear as a vacuum configuration but can only appear as the mostprobable value in the coherent state.The above discussion then raises the following question. How about choosing deltafunction states for any k in Ψ k given in (2.7)? Clearly the delta function states in the con-figuration space would appear to serve as better candidates because they would exactlyreproduce the background (2.4) with probability 1. Unfortunately however, this configu-ration doesn’t survive long and the delta function states expand very fast in the presenceof the interacting M-theory Hamiltonian. This ruins our hope of realizing (2.4) (or equiva-lently (2.2) in IIB) purely as a classical solution with zero quantum width, reinforcing, yetagain, the coherent state nature of the background.The coherent state description for any given mode k for a given Schr¨odinger wave-function in (2.7) is clearly the best possible description we can have for our case. Let usnow see how we can quantify this further. For simplicity we will look at one specific mode, (cid:101) g µν ( k ), with the spatial part of the Schr¨odinger wave-function ψ k ≡ ψ k ( x , y, z ). The Fouriermode decomposition (2.9) will tell us precisely the most probable amplitude for (cid:101) g µν ( k ), andlet us denote it by α ( ψ ) µν ( k , t ). The temporal behavior of α ( ψ ) µν ( k , t ) is clearly oscillatory withfrequency ω ( ψ ) k , which is the same ω ( a ) k that appeared earlier in our discussion. The subtletywith the off-shell parts of (2.6) will be dealt soon. Meanwhile we define: α ( ψ ) µν ( k , t ) ≡ α ( ψ ) µν ( k , ω k )exp (cid:16) − iω ( ψ ) k t (cid:17) , (2.12)where α ( ψ ) µν ( k , ω k ) can be defined up to a possible phase of exp (cid:16) iσ ( ψ ) k (cid:17) . Collecting every-thing together then provides the following wave-function in the configuration phase for themode (cid:101) g µν ( k ): Ψ (cid:104) α ( ψ ) µν ( k ,t ) (cid:105) ( (cid:101) g µν ( k ) , t ) = (cid:32) ω ( ψ ) k π (cid:33) / exp − ω ( ψ ) k (cid:32)(cid:101) g µν ( k ) − ω ( ψ ) k Re (cid:104) α ( ψ ) µν ( k , t ) (cid:105)(cid:33) × exp (cid:32) i Im (cid:104) α ( ψ ) µν ( k , t ) (cid:105) (cid:101) g µν ( k ) + iθ ( ψ ) k ( t ) (cid:33) , (2.13) One needs to be careful to not interpret these delta function states as localized states in space-time. Inspace-time we will continue to have the standard Schr¨odinger wave-functions Ψ k ( x , y, z, t ) (2.7). Here, ina similar vein as with the realization of the coherent states in the configuration space, the delta functionstates will also be realized in the configuration space. – 20 –here no sum over repeated indices are implied above; and θ ( ψ ) k ( t ) is yet another phasethat appears in the definition of the wave-function for the mode (cid:101) g µν ( k ). This phase can bedetermined by demanding that Ψ (cid:104) α ( ψ ) µν ( k ,t ) (cid:105) ( (cid:101) g µν ( k ) , t ) solves the harmonic oscillator wave-equation that appears from the interacting Lagrangian in the M-theory picture. As dis-cussed earlier, the interactions are essential and they make the Schr¨odinger wave-function in (2.13) simple but render the wave-functions in (2.7) rather complicated. Neverthelessthe form for Ψ (cid:104) α ( ψ ) µν ( k ,t ) (cid:105) ( (cid:101) g µν ( k ) , t ) may be determined precisely even if ψ k may be involvedas we saw above. The phase θ ( ψ ) k ( t ) then becomes: θ ( ψ ) k ( t ) = − (cid:104) ω ( ψ ) k t − | α ( ψ ) µν ( k , | sin (cid:16) ω ( ψ ) k t − σ ( ψ ) k (cid:17)(cid:105) , (2.14)where σ ( ψ ) k is the initial phase of the eigenvalue α ( ψ ) µν ( k ,
0) discussed above. From (2.13),it is easy to infer that (cid:12)(cid:12)(cid:12)(cid:12) Ψ (cid:104) α ( ψ ) µν ( k ,t ) (cid:105) ( (cid:101) g µν ( k ) , t ) (cid:12)(cid:12)(cid:12)(cid:12) peaks exactly at Re (cid:104) α ( ψ ) µν ( k , t ) (cid:105) , which inturn oscillates as sin (cid:16) ω ( ψ ) k t (cid:17) , as one would expect. On the other hand, a wave-function ofthe form: Ψ (cid:104) α ( ψ ) µν ( k , (cid:105) ( (cid:101) g µν ( k ) ,
0) = (cid:89) k δ (cid:16)(cid:101) g µν ( k ) − α ( ψ ) µν ( k , (cid:17) , (2.15)defined specifically at t = 0, would reproduce the background (2.4) in the exact classicalform with zero quantum width at t = 0. However as discussed above, this immediatelyexpands to become highly quantum with no trace of the classical picture left resemblinganything close to (2.4).Interestingly, the wave-function (2.13), itself has a gaussian form with the center of thegaussian behaving in a simple harmonic oscillator fashion, up-to an overall phase factor.This phase factor can be simplified a bit if α ( ψ ) µν ( k , t ) becomes real, but not too much. Theoverall phase of exp (cid:16) iθ ( ψ ) k ( t ) (cid:17) given in (2.14) cannot be made to vanish, and will survive.The probablity however remains peaked at the desired value. In terms of Schr¨odinger operator formalism this implies: (cid:104) δ g µν ( x, y, z ) (cid:105) α ( ψ ) = Re (cid:32)(cid:90) d k ω ( ψ ) k α ( ψ ) µν ( k , t ) ψ k ( x , y, z ) (cid:33) , (2.16)where the expectation value of the fluctuation of metric operator, δ g µν ( x, y, z ) is taken overthe coherent state (2.13). Expectedly the integral is done over d k because ω ( ψ ) k appearingfrom (2.12) is related to k on-shell . The puzzle however is the off-shell part from (2.6).What would be the simplest way of reproducing that? This is in general tricky, so let us To remind the readers again, there are at least two set of Schr¨odinger equations that can appear here.The Schr¨odinger equations whose solutions are the four set of wave-functions in (2.7) are never of thesimple harmonic oscillator form. In fact these Schr¨odinger equations allow highly non-trivial potentials(for example (1.5)). On the other hand, the Schr¨odinger equations allowing wave-functions like (2.13) havesimple harmonic oscillator potentials. – 21 –ropose the following expression for α ( ψ ) µν ( k , t ): α ( ψ ) µν ( k , t ) ≡ (cid:34) α (1; ψ ) µν ( k , ω ( ψ ) k ) + lim ω o → ω ( ψ ) k α (2; ψ ) µν ( k ) | ω o |√ π e − (cid:16) ω ( ψ ) k /ω o (cid:17) (cid:35) e − iω ( ψ ) k t , (2.17)where the first term, α (1; ψ ) µν ( k , ω ( ψ ) k ) is the on-shell piece and the second term, α (2; ψ ) µν ( k ), isrelated to the off-shell piece. The extra factor of 2 ω ( ψ ) k is essential to eliminate the measureof the integral (2.16), with the limiting value of ω o → δ ( ω ( ψ ) k ). One advantage of such an approach is that we can continue using the on-shell integral form (2.16) to express the metric components even if there are off-shell piecesin the Fourier transforms. The disadvantage however is that the inverse Fourier transformis a bit tricky: one will have to resort back to d k to get all the factors correctly.Let us make a few more observations. First, the way we have expressed (2.17), tells usthat the off-shell piece is in general arbitrarily small, but does appear to provide non-zerovalue once integrated over an on-shell integral of the form (2.16). This is good because it’dmean that we don’t have to worry too much about the off-shell parts from (2.6) and continueusing our on-shell analysis. Such a point of view will become very useful when we have toextract values from path-integral computations in section 2.4. Of course there does existother ways to deal with the off-shell parts, but here we will pursue the simplest one. Second,we haven’t added the solitonic background η µν h − / ( y, x ) to it. We could in principle addthis to (2.16) using the completeness condition of the coherent states. This way the fullon-shell metric configuration (2.4) appears from our coherent state description. On theother hand, in terms of Feynman field formalism, this on-shell relation is not required (infact fields are maximally off-shell ), and therefore we expect: g µν ( x, y, z ) = η µν h / ( y, x ) + (cid:90) d k (cid:101) g µν ( k , k ) ψ k ( x , y, z ) e − ik t , (2.18)where with some abuse of notation we used (cid:101) g µν ( k , k ) to denote the field amplitudes .Thus here, (cid:101) g µν ( k , k ) is generic and should not be confused with the Fourier decomposition(2.9). As expected, there is also no relation between k and k , and the integral spans overthe full eleven-dimensional momentum space.There is an immediate advantage of expressing the fields in terms of the off-shell form(2.18), that is not there in the on-shell form (2.16). The form (2.18) allows us to study thedynamics of the system more consistently than the evolution of the most probable statein (2.16). The expression (2.18) captures any quantum width of a state, no matter howsharply peaked the configuration space wave-functions are. In fact for a state like (2.15),where any field configuration would go wildly off-shell, (2.18) is well-suited to tackle thedynamics. Additionally, the time-independencies of the modes ψ k ( x , y, z ), are essential tostudy the Wilsonian effective action that formed the basis of our analysis in [21] and [22]. In the operator formalism, where g µν ( x, y, z ) becomes an operator, then can be expandedin terms of the corresponding creation and annihilation operators for the modes ψ k ( x , y, z ). Needless to say, the notations g µν will henceforth denote metric operator and g µν the field. – 22 –t is instructive to present this formalism and compare the result in terms of the modesexpanded around de Sitter vacuum . First let us express the modes over the solitonicbackground (2.1). This takes the form: g µν ( x , y, z ; t ) − η µν h / ( y, x ) = (cid:90) d k (2 π ) (cid:113) ω ( ψ ) k (cid:88) s = ± (cid:104) a s ( k ) e µν ( k , s ) ψ k ( x , y, z )exp (cid:16) − iω ( ψ ) k t (cid:17) + a † s ( k ) e ∗ µν ( k , s ) ψ ∗ k ( x , y, z )exp (cid:16) iω ( ψ ) k t (cid:17) (cid:105) , (2.19) where e µν ( k , s ) is the polarization tensor for a given momentum k and s = ± ; with( a s ( k ) , a † s ( k )) forming the standard annihilation and the creation operators. Note howeverthe difference from usual QFT mode expansion: the spatial modes are not simple and theyare given in terms of the Schr¨odinger wave-functions ψ k ( x , y, z ), whereas the temporalmodes take the usual form of exp (cid:16) ± iω ( ψ ) k t (cid:17) with t being the time coordinate used here.The integral in (2.19) is over d k and not over d k as one would expect and we haveexpressed the temporal behavior using the on-shell value of ω ( ψ ) k . We could have used alsothe off-shell form exp( ± ik t ), but then we will have to specify the pole at ω ( ψ ) k . Of course,this is all very standard, so we won’t elaborate it further.We will however compare the above mode expansion (2.19), which is over the solitonicbackground (2.1), to the one over the uplifted background (2.4). The mode expansion nowtakes the following form : g µν ( x , y, z ; t ) − η µν (cid:16) Λ | t | (cid:112) h ( y ) (cid:17) / = (cid:90) d k (2 π ) (cid:113) ω ( ψ ) k (cid:88) s = ± (cid:104) ˆ a s ( k ) e µν ( k , s ) ˆ ψ k ( x , y, z )exp (cid:16) − i ˆ ω ( ψ ) k ( t ) t (cid:17) + ˆ a † s ( k ) e ∗ µν ( k , s ) ˆ ψ ∗ k ( x , y, z )exp (cid:16) i ˆ ω ( ψ ) k ( t ) t (cid:17) (cid:105) , (2.20) where as before e µν ( k , s ) denotes the polarization tensor, but now there are quite a fewnoticeable differences. The spatial modes ˆ ψ k ( x , y, z ) are more involved than the spatialmodes ψ k ( x , y, z ) encountered earlier. This is expected, because the background (2.4) isdifferent from the solitonic background (2.1). A more crucial thing is the appearance ofboth ˆ ω ( ψ ) k ( t ) and ω ( ψ ) k in (2.20). In fact ˆ ω ( ψ ) k ( t ) is a much more complicated function of k and t , and we expect: ˆ ω ( ψ ) k ( t ) = ω ( ψ ) k + it log (cid:104) g ( ψ ) ( k , t ) (cid:105) , (2.21)where the second term implies that the frequencies themselves are time-dependent. Infact we are not restricted to real values now and g ( ψ ) ( k , t ) can be imaginary, leading toˆ ω ( ψ ) k ( t ) becoming imaginary, although ω ( ψ ) k remains real throughout. This difference is A word of caution here. The metric has 66 degrees of freedom, and in the presence of other fields,namely the G-fluxes (we will discuss them soon), some of these degrees of freedom have to be gauged. Thisgauging sometimes lead to propagating ghosts, but one may choose a gauge choice where the propagatingghosts are absent (this is easy to achieve either in the effective four-dimensional case in type IIB or in theeffective three-dimensional case in M-theory with only metric degrees of freedom). Subtlety appears whenboth metric and fluxes are present, but we will not worry about them right now, and consider them onlyin section 3.3. – 23 –mportant and spells out the fact that the frequency of a given mode can change withtime, both in terms of the modulus and argument of a complex number, leading to allkinds of Trans-Planckian issues encountered in the literature (see for example [25]).One may quantify the above mode expansion directly from an effective 2+1 dimensionalpoint of view. In such an effective picture , the internal degrees of freedom at least to firstapproximation do not effect the mode expansions, implying that ˆ ψ k ( x , y, z ) = exp( i k · x ).This could also be interpreted as though we have taken a slice of the internal manifoldwith fixed ( y, z ). The mode expansion over the 2 + 1 dimensional space-time (2.4) can beexpressed as in (2.20) with fixed ( y, z ), such that:ˆ ω ( ψ ) k = it log (cid:104) | k | t / (cid:16) c J / ( | k | t ) + c Y / ( | k | t ) (cid:17)(cid:105) , (2.22)where | k | = √ k · k ; and J n ( | k | t ) and Y n ( | k | t ) are the Bessel functions of the first and thesecond kinds respectively. The c i are constants, but they cannot both be real, becausewe want to extract a factor of exp( − i | k | t ) from (2.22). Their precise value can be easilydetermined by demanding that (2.22) takes the form (2.21). Note that the dimensionsare taken care of inside the logarithm by introducing appropriate powers of the Hubbleconstant H. The exact form for (2.22) is not important, but what is important howeveris to note that (2.22) implies time-dependent frequency for the modes expanded over theup-lifted de sitter background (2.4) in M-theory. Dimensionally reducing to IIB, which isthe same as expanding over the four-dimensional part of (2.2), the fluctuating modes havea frequency given by:ˆ ω ( ψ ) k = it log (cid:26) | k | (cid:104) c (cid:16) sin | k | t − | k | t cos | k | t (cid:17) + c (cid:16) | k | t sin | k | t + cos | k | t (cid:17)(cid:105)(cid:27) (2.23)which although expectedly differs from (2.22), carries the same information as above.Again, the c i constants cannot be all real, and comparing to (2.21), it is easy to inferthat c = i and c = −
1. This then leads to the familiar result in the literature with g ( ψ ) ( k , t ) in (2.21) taking the form: g ( ψ ) ( k , t ) = (cid:115) t + 1 | k | exp (cid:16) i tan − ( | k | t ) (cid:17) , (2.24)where one would have to again insert the Hubble constant H to make the quantity underthe square-root to have the right dimension (to avoid clutter, we will henceforth take H = 1unless mentioned otherwise). Both the results, (2.22) and (2.24), show that the frequenciesin M-theory and IIB respectively are time varying frequencies, thus would in principlecreate problems if we try to integrate out the high energy modes in the Wilsonian way.Such an issue do not exist when we express the de Sitter space as a coherent state becausethe modes are described using (2.19). In fact any fluctuations, even the ones that couldbe interpreted as over the coherent state themselves, should be expressed using the modesover the solitonic vacuum (2.1). For example the case with C ( p ) nm << – 24 –et us quantify the above statement more carefully. The effective fluctuation spectraover a de Sitter vacuum in IIB or its uplift in M-theory typically go like exp (cid:16) i k · x − i ˆ ω ( ψ ) k ( t ) t (cid:17) , as we saw above. Our concern is with the frequencies of the modes as theydepend on the conformal time t . Our goal would be to express this as a linear combinationof the modes over the solitonic vacuum, either in M-theory or in IIB. In other words, weexpect: exp (cid:16) − i ˆ ω ( ψ ) k ( t ) t (cid:17) = (cid:90) dk f ( k , k ) exp ( − ik t ) , (2.25)where the RHS is expressed with modes over the solitonic vacuum. At this stage we don’tcare whether the modes k take the on-shell values ω k , and typically this may not alwaysbe possible. For the modes taking the form given in (2.21), the Fourier coefficients f ( k , k )can be expressed in the following way: f ( k , k ) = (cid:90) + T − T dt exp (cid:20) − i (cid:16) ω ( ψ ) k + it − log (cid:0) g ( ψ ) ( k , t ) (cid:1) − k (cid:17) t (cid:21) , (2.26)where g ( ψ ) ( k , t ) appears in (2.21), and T (or more appropriately √ Λ T ) denotes temporalboundary. The Fourier coefficient f ( k , k ) is an effective way to interpret the fluctuationsover a coherent state, but the problem with an expression like (2.26) is that it may notalways be convergent . The issue of convergence stems from the fluctuating modes them-selves. For example the temporally varying frequencies in IIB as given in (2.23) tend toblow-up as | k | t → ±∞ , and this is reflected directly in the Fourier coefficient f ( k , k ) oncewe plug-in (2.24) in (2.26) to get: f ( k , k ) = 1 | k | δ ( | k | − k ) − | k | − k T cos (cid:104) ( | k | − k ) T (cid:105) + 2( | k | − k ) sin (cid:104) ( | k | − k ) T (cid:105) , (2.27) where the expected non-convergence appears from the T dependence of the second term in(2.27). This may not quite be an issue because we could keep T large but not necessarilyinfinite (a careful study with path integrals in section 2.5 will show that this problemdoes not arise). Note also that the way we have represented f ( k , k ), it is a real functionso the standard relation between f ∗ ( k , k ) and f ( − k , ± k ) do not hold here because thetransformation (2.25) is arranged to reproduce a complex function, not a real one. In asimilar vein, in M-theory, one will have to reproduce (2.22) using the Fourier transform(2.25). The issue of convergence appears here too, as both x / J / ( x ) and x / Y / ( x ) in(2.22) blow-up for x ≡ | k | t → ∞ . As before we can bound T to a large but finite value sothat an expression like (2.27), but now for M-theory, makes sense.Thus it appears from an effective d + 1 dimensional point of view ( d = 2 for M-theoryand d = 3 for IIB), the fluctuations over de Sitter vacuum , can be represented here as alinear combination of the modes over the solitonic vacuum. In other words: δg µν ( x ) = (cid:88) s = ± (cid:90) d d +1 k e µν ( k , s ) f ( k , k ) ψ k ( x ) e ik t (2.28) This is an abuse of notation. The dimensionless time parameter is always √ Λ t , where Λ is the cosmo-logical constant. – 25 – (cid:88) s = ± (cid:90) d d k e µν ( k , s ) ψ k ( x ) (cid:90) dk f ( k , k ) e ik t ≡ (cid:88) s = ± (cid:90) d d k e µν ( k , s ) f k ( t ) ψ k ( x ) , where ψ k ( x ) = (cid:82) d D − d − k ψ k ( x , y , z ) for D space-time dimensions; and with some abuseof notations we have used k to also signify the momenta along the internal directions.Which is which, should be clear from the context.The point of the above exercise (2.28) is simple. It is to show that the fluctuationsmay be controlled by a time varying amplitude for any d -dimensional momentum k , andthus takes the expected standard form. In fact demanding reality of δg µν ( x ), reproducesthe familiar constraint: f ∗ k ( t ) = ( − d +1 f − k ( t ) with real e µν ( k , s ) and ψ ∗ k ( x ) = ψ − k ( x ).Combining all these we can then perform the following series of manipulations from (2.28): δg µν ( x ) = 12 (cid:88) s = ± (cid:90) + ∞−∞ d d k e µν ( k , s ) (cid:20)(cid:18) f k + b ∂∂f ∗ k + f k − b ∂∂f ∗ k (cid:19) ψ k ( x ) (cid:21) (2.29)= 12 (cid:88) s = ± (cid:90) + ∞−∞ d d k e µν ( k , s ) (cid:20)(cid:18) f ∗ k + b ∂∂f k (cid:19) ψ − k ( x ) + (cid:18) f k − b ∂∂f ∗ k (cid:19) ψ k ( x ) (cid:21) , where in the first line we simply added and subtracted a derivative piece, but in the secondline a redefinition of the variable k puts it in a much more suggestive format. The quantity b appearing in (2.29) is related to the ground state wave-function of the configurationspace (not to be confused with the wave-functions in spacetime!), in the following way:Ψ ( f k ) ≡ (cid:104) f k | (cid:105) = N k exp (cid:18) − | f k | b (cid:19) , (2.30)with N k forming the normalization constant, and b provides the Gaussian width of theground state. Note that this wave-function is only for the mode k , and the generic wave-function for the ground state | (cid:105) is a matrix product of the wave-functions of the form (2.30).Clearly shifting (2.30) in the configuration space should give us the required coherent statefor the mode k , so the question is whether we can quantify the shift in a precise way .This is where the decomposition (2.29) pays off, because: (cid:18) f ∗ k + b ∂∂f k (cid:19) Ψ ( f k ) = 0 (cid:18) f k − b ∂∂f ∗ k (cid:19) Ψ ( f k ) = 2 N k f k exp (cid:18) − | f k | b (cid:19) , (2.31)implying that the first operator annihilates the vacuum wave-function whereas the secondoperator creates a new wave-function. The new wave-function is exactly proportional tothe first excited state of a harmonic oscillator, implying that the second operator acts as a creation operator! These are then the Schr¨odinger representations of the familiar creationand the annihilation operators. Thus we can identify: a k ≡ (cid:115) ω ( ψ ) k (cid:18) f ∗ k + b ∂∂f k (cid:19) , a † k ≡ (cid:115) ω ( ψ ) k (cid:18) f k − b ∂∂f ∗ k (cid:19) , (2.32) A more precise identification of the coherent states to the shifted interacting vacuum will be discussedin section 2.4. Meanwhile what we have here should suffice. – 26 –onnecting us with the mode expansion (2.19) proposed earlier provided we identify a k and a † k from (2.32) with a s ( k ) and a † s ( k ) respectively from (2.20), and go to the Heisenbergpicture. The spin s informations in the definitions of the creation and the annihilationoperators in (2.20) are redundant because for either choices of s in (2.20) the creation orthe annihilation operators remain unchanged (so we will ignore them in the subsequentdiscussion). Note that imposing [ a k , a † k (cid:48) ] = δ ( k − k (cid:48) ) makes b = (cid:16) ω ( ψ ) k (cid:17) − / , which isconsistent from (2.13). Finally the effective space-time wave-function for a given modeappears naturally from the one-point function (treating δ g µν ( x, s ) as an operator): (cid:104) | δ g µν ( x, s ) | k (cid:105) = e µν ( k , s ) ψ − k ( x ) exp (cid:16) iω ( ψ ) k t (cid:17) , (2.33)which is as one would expect for any fluctuation over a solitonic background providedwe identify ψ − k ( x ) = ψ ∗ k ( x ). This again confirms the fact that modes over a solitonicbackground may have non-trivial spatial wave-functions , if h = h ( y, x ) in (2.1), but thetemporal dependences remain simple with time-independent frequencies.On the other hand, in the same space-time, i.e. over the solitonic background (2.1),we can construct fluctuations (2.28) that could have different interpretations. For examplea kind of fluctuation in eleven dimensional space-time that we want to reproduce wouldbe: (cid:104) δ g µν ( x, y, z ) (cid:105) Ψ ( ψ ) = (cid:88) s = ± (cid:90) d k (2 π ) b ( ψ ) k e µν ( k , s ) ˆ ψ k ( x , y, z ) exp (cid:16) − i ˆ ω ( ψ ) k ( t ) t (cid:17) (2.34)+ (cid:88) s = ± (cid:90) d k (2 π ) c ( ψ ) k e ∗ µν ( k , s ) ˆ ψ − k ( x , y, z ) exp (cid:16) + i ˆ ω ( ψ ) k ( t ) t (cid:17) , where the subscript Ψ ( ψ ) denotes some state over the solitonic vacuum (2.1); and ( b ( ψ ) k , c ( ψ ) k )are coefficients that only depend on k . As we saw before for the effective case in (2.28),fluctuations on the RHS of (2.34) could be achieved with a choice of f ( k , k ) taking theform (2.26). However in quantum theory, it is more important to find a state Ψ ( ψ ) thatreproduces the fluctuations as an expectation value over the state itself. Question then is:what would be the form of Ψ ( ψ ) ? From the RHS of (2.34), one might presume Ψ ( ψ ) , for agiven mode k , to be a state of the following form: (cid:12)(cid:12)(cid:12) Ψ ( ψ ) k ( t ) (cid:69) = (cid:88) n c ( ψ ) n ( k , t ) exp (cid:20) − i (cid:18) n + 12 (cid:19) ω ( ψ ) k t (cid:21) | n ; k , ψ k (cid:105) , (2.35)in the configuration space, expressed in terms of a linear combinations of the eigenstateswith time-dependent coefficients. This extra time-dependence is necessary because theexpectation value (2.34) involve mode expansions from (2.19) that could only relate oneup or one down states in the configuration space. Thus only a linear combination ofeigenstates for a given momentum k with time-dependent coefficients could provide thetemporal behavior with frequencies ˆ ω ( ψ ) k ( t ) as in (2.21). However we will also need toworry about an overlap integral of the form: (cid:90) d x ψ ∓ k ( x , y, z ) ˆ ψ ± k (cid:48) ( x , y, z ) h − ( x , y ) h / ( y ) ≡ r ( k , k (cid:48) ) , (2.36)– 27 –etween the spatial wave-function ψ k ( x , y, z ) over the solitonic background (2.1) and thespatial wave-function ˆ ψ k (cid:48) ( x , y, z ) over the M-theory uplifted background (2.4). This overlapcondition should in turn be compared to the orthogonality condition that appeared infootnote 5. The question now is whether we can quantify the function r ( k , k (cid:48) ). For this,note that at any given time √ Λ | t | ≡ T , the uplifted metric (2.4) resembles the solitonicbackground (2.1) by some redefinitions of the coordinates by constant factors, implying thatthe wave-function ˆ ψ k ( x , y, z ) cannot be very different from ψ k ( x , y, z ). In other words, wecan expect: ˆ ψ k ( x , y, z ) = (cid:90) d k (cid:48) r ( k , k (cid:48) ) ψ k (cid:48) ( x , y, z ) ≈ (cid:88) k (cid:48) r kk (cid:48) ψ k (cid:48) ( x , y, z ) , (2.37)where in the second equality we have assumed the wave-functions are discrete ( i.e. in abox). Our discussions above will tell us that the function r ( k , k (cid:48) ) is sharply peaked near k (cid:48) = k , implying that the overlap is sub-leading when k (cid:48) (cid:54) = k , in other words r ( k , k (cid:48) ) ≈ r k δ ( k − k (cid:48) ). This further means that c ( ψ ) n ( k , t ) coefficients in (2.35) are related by thefollowing set of equations, the first one being : (cid:113) ω ( ψ ) k b ( ψ ) k exp (cid:16) iω ( ψ ) k t (cid:17) g ( ψ ) ( k , t ) = (cid:88) n √ n r ( k , t ) c ( ψ ) n ( k , t ) c ∗ ( ψ ) n − ( k , t ) (2.38) ≡ (cid:88) l d ( ψ ) l ( k ) exp (cid:20) − i (cid:18) l + 12 (cid:19) ω ( ψ ) k t (cid:21) , where g ( ψ ) ( k , t ), as defined in (2.21), could in principle be a complex number; and r ( k , t ) = r k (cid:80) n | c ( ψ ) n ( k ,t ) | . Note that in the second line of (2.38), we have expressed the sum of the Note that for the generic relation, without imposing any specific condition on the overlap function r ( k , k (cid:48) ), one has to start with the following state: (cid:12)(cid:12)(cid:12) Ψ ( ψ ) ( t ) (cid:69) ≡ (cid:12)(cid:12)(cid:12) Ψ ( ψ ) k ( t ) (cid:69) ⊗ (cid:12)(cid:12)(cid:12) Ψ ( ψ ) k ( t ) (cid:69) .... ≡ ⊗ k (cid:12)(cid:12)(cid:12) Ψ ( ψ ) k ( t ) (cid:69) which is constructed out of the product of all allowed momenta k . Here we take them as discrete to givesome meaning to the otherwise infinite product of states labelled by the continuous variable k . Using thisas the input on the LHS of (2.34), and comparing the terms proportional to e µν ( k , s ), gives us the followingrelation: b ( ψ ) k exp (cid:16) iω ( ψ ) k t (cid:17) g ( ψ ) ( k , t ) = Z ( t ) (cid:90) d k (cid:48) (cid:113) ω ( ψ ) k (cid:48) (cid:80) n √ n r ( k , k (cid:48) ) c ( ψ ) n ( k (cid:48) ,t ) c ∗ ( ψ ) n − ( k (cid:48) ,t ) (cid:80) n (cid:12)(cid:12)(cid:12) c ( ψ ) n ( k (cid:48) ,t ) (cid:12)(cid:12)(cid:12) which boils down to the first equality in (2.38) when r ( k , k (cid:48) ) ≈ r k δ ( k − k (cid:48) ), upto the normalization factorof Z . This factor is defined as: Z ( t ) ≡ exp (cid:34)(cid:90) d k log (cid:32)(cid:88) n (cid:12)(cid:12)(cid:12) c ( ψ ) n ( k , t ) (cid:12)(cid:12)(cid:12) (cid:33)(cid:35) which would not appear if we normalize the overall state (cid:12)(cid:12)(cid:12) Ψ ( ψ ) ( t ) (cid:69) from the start itself. Interestingly thisnormalization factor is time-dependent, and so is (cid:80) n | c ( ψ ) n ( k , t ) | . We will assume such normalization isdefined even for the state (2.35), so that with the delta function choice for r ( k , k (cid:48) ) we can reproduce (2.38)without any extra factors. Note however that the second equality in (2.38) is not much effected by choosinga generic form of r ( k , k (cid:48) ). – 28 –roducts of the c ( ψ ) n ( k , t ) coefficients as another linear combination expressed in terms oftime-independent coefficients. In a similar vein, the other set becomes: (cid:113) ω ( ψ ) k c ( ψ ) k exp (cid:16) − iω ( ψ ) k t (cid:17) (cid:104) g ( ψ ) ( k , t ) (cid:105) − = (cid:88) n √ n + 1 r ( k , t ) c ( ψ ) n ( k , t ) c ∗ ( ψ ) n +1 ( k , t ) ≡ (cid:88) l e ( ψ ) l ( k ) exp (cid:20) − i (cid:18) l + 12 (cid:19) ω ( ψ ) k t (cid:21) , (2.39)where again we have expressed the sum of the products of c ( ψ ) n ( k , t ) coefficients in terms ofa series defined with coefficients e ( ψ ) l . These coefficients are not hard to find, and one canshow that: d ( ψ ) l ( k ) ≡ √ | ω ( ψ ) k | / b ( ψ ) k (cid:90) + ∞−∞ dt g ( ψ ) ( k , t ) exp (cid:20) i (cid:18) l + 32 (cid:19) ω ( ψ ) k t (cid:21) e ( ψ ) l ( k ) ≡ √ | ω ( ψ ) k | / c ( ψ ) k (cid:90) + ∞−∞ dt (cid:104) g ( ψ ) ( k , t ) (cid:105) − exp (cid:20) i (cid:18) l − (cid:19) ω ( ψ ) k t (cid:21) , (2.40)where the modulus keeps only the positive frequencies ω ( ψ ) k ; with g ( ψ ) ( k , t ) as in (2.21) and l ∈ Z . Note, because of the 3 / / d ( ψ ) l ( k ) and e ( ψ ) l ( k ) are different functionsof the momenta k . In IIB the only changes would be d k instead of d k in (2.34), and g ( ψ ) ( k , t ) taking the form (2.24) instead of the one derived from (2.22). For example inIIB, d ( ψ ) l ( k ) becomes: d ( ψ ) l ( k ) = √ b ( ψ ) k (cid:12)(cid:12) l + (cid:12)(cid:12) (cid:40) δ ( k ) − T cos (cid:20)(cid:18) l + 32 (cid:19) k T (cid:21) + 2 sin (cid:2)(cid:0) l + (cid:1) k T (cid:3)(cid:12)(cid:12) l + (cid:12)(cid:12) k (cid:41) , (2.41)where we have defined ω ( ψ ) k = k , and T as before is to be taken to be very large, butnot infinite. Expectedly the T behavior is similar to what we saw earlier in (2.27). Notehowever that the first term fixes k to k = 0. Thus they are zero momentum states witharbitrary energy, so will appear as off-shell states.There is yet another condition in addition to (2.38) and (2.39), which has to do withcertain orthogonality relations between the states (2.13) and (2.35). To quantify this, let usexpress the wave-function (2.13) as (cid:68)(cid:101) g µν ( k ) (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) , where the ket would correspond tothe coherent state for momentum k . They are not orthogonal states, in addition to beingover-complete, but we want to demand at least the following orthogonality conditions: (cid:68) Ψ ( ψ ) k (cid:48) ( t ) (cid:12)(cid:12)(cid:12) δ g µν ( x, y, z ) (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) = (cid:68) Ψ ( α ) k (cid:48) ( t ) (cid:12)(cid:12)(cid:12) δ g µν ( x, y, z ) (cid:12)(cid:12)(cid:12) Ψ ( ψ ) k ( t ) (cid:69) = 0 , (2.42)for all momenta k and for all time. The above orthogonality is a bit harder to achieve inthe light of the two additional constraints (2.38) and (2.39), but is not impossible giventhat the number of c ( ψ ) n ( k , t ) coefficients are infinite in the definition of the state (2.35).In fact since the ket (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) can be expressed as a linear combinations of eigen-states,– 29 –2.42) imposes two linear relations between the coefficients c ( ψ ) n ( k , t ), which become onewhen the operator δ g µν ( x, y, z ) becomes real. In either case then, a generic state of theform: (cid:12)(cid:12)(cid:12) Ψ ( c c ) k ( t ) (cid:69) ≡ c (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) + c (cid:12)(cid:12)(cid:12) Ψ ( ψ ) k ( t ) (cid:69) , (2.43)where ( c , c ), which are constants independent of ( k , t ), would succinctly capture all theinformation that we want once we take an expectation value of the operator δ g µν ( x, y, z )over (2.43) and add the background solitonic value in the following way: η µν h ( x , y ) + 1 N (cid:90) d k (cid:68) Ψ ( c c ) k ( t ) (cid:12)(cid:12)(cid:12) δ g µν ( x, y, z ) (cid:12)(cid:12)(cid:12) Ψ ( c c ) k ( t ) (cid:69) , (2.44)where c = 1 , c = 0 reproduces the classical de Sitter background and c = 1 , c (cid:54) =0 reproduces the fluctuation spectra over the classical de Sitter background with N = (cid:104) Ψ ( c c ) k ( t ) (cid:12)(cid:12) Ψ ( c c ) k ( t ) (cid:105) . The miraculous thing is that all these are over the solitonic back-ground (2.1) and we are able to reproduce the de Sitter results as expectation values.The other metric modes of the theory, namely g mn , g αβ and g ab would also be describedusing coherent state wave-functions of the form (2.13), although specific details of the con-structions might differ. The interactions that are required to construct the fluctuationwave-functions ( η k , ξ k , ζ k ), i.e. the solutions of the corresponding Schr¨odinger equations,are necessarily with the soliton themselves and all interactions between the modes (includ-ing self-interactions) go in the definition of the interacting Hamiltonian. This interactingHamiltonian becomes more complicated once the UV degrees of freedom are further inte-grated out. One could also spell out equivalent operator and Feynman prescription as in(2.16) and (2.18) respectively. These description do become simpler from 2 + 1 dimensionalperspective because the internal metric appear as scalar fields there. We will come back tothis a bit later. Let us take this opportunity to discuss two related topics, one, dealing with the number ofgravitons in a coherent state and two, dealing with the excitation of a coherent state. Thesecond case, i.e. the one related to excited coherent states, is a rich subject in itself andbasically deals with the dynamics of a coherent state once we add m number of gravitons.This was originally developed for the photon case by Agarwal and Tara [36], and unfortu-nately here we will only be able to elaborate the bare minimum required for our purpose.Interested readers may want to go to the original papers in the subject starting with [36].The question that we want to ask here is what happens when we fluctuate the coher-ent state, with wave-function Ψ (cid:104) α ( ψ ) µν ( k ,t ) (cid:105) ( (cid:101) g µν ( k ) , t ) as in (2.13), by adding m number ofgravitons of momenta k . The goal of the exercise is to see whether there is any tangi-ble connection between graviton-added coherent states (GACS) and the state (cid:12)(cid:12)(cid:12) Ψ ( c c ) k ( t ) (cid:69) constructed in (2.43). Note that the states (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) and (cid:12)(cid:12)(cid:12) Ψ ( ψ ) k ( t ) (cid:69) are not necessarily orthogonal to each other. Additionallytwo coherent states (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) and (cid:12)(cid:12)(cid:12) Ψ ( β ) k ( t ) (cid:69) are also not orthogonal to each other unless (cid:12)(cid:12)(cid:12) α ( ψ ) k − β ( ψ ) k (cid:12)(cid:12)(cid:12) (cid:29) – 30 –o proceed, certain redefinitions of the coordinates in the configuration space mightease our computations. For example in the configuration space wave-function (2.13), we canredefine the Fourier modes (cid:101) g µν ( k ) and (cid:101) α µν ( k , t ) as e µν f k and e µν α ( ψ ) k ( t ) respectively withthe condition e µν e µν ≡ f k appearinghere is the same f k that appears in the definition of the creation and the annihilationoperators in (2.32). Thus our way of expressing the Fourier modes would then be generic.However the condition on the polarization tensor is not generic and there exists other waysto fix the product, but for our purpose we will stick to the simplest case here. With thesedefinitions, the form of our wave-function (2.13) simplifies from: Ψ (cid:104) α ( ψ ) µν ( k ,t ) (cid:105) ( (cid:101) g µν ( k ) , t ) ≡ (cid:68)(cid:101) g µν ( k ) (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) −→ Ψ ( α ) k ( f k , t ) ≡ (cid:68) f k (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) , (2.45)implying that the coherent state that control the dynamics for any momentum k and anyinstant of time t is (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) . Question we want to ask is what happens when the coherentstate is acted on by an operator of the form: G ( ψ ) (cid:16) a k + a † k ; t (cid:17) ≡ ∞ (cid:88) n =0 C ( ψ ) n k ( t ) (cid:16) z a k + z a † k (cid:17) n , (2.46)where C ( ψ ) n k ( t ) are generic time-dependent coefficients and z i are time-independent con-stants. Note that this operation will lead to a state more generic than the usual Agarwal-Tara [36] type state , but more importantly the time-dependent coefficients in (2.46) mighttie up with what we discussed in the previous section. The state then becomes: (cid:12)(cid:12)(cid:12) Ψ ( αg ) k (cid:69) ≡ G ( ψ ) (cid:16) a k + a † k ; t (cid:17) (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) = ∞ (cid:88) n =0 C ( ψ ) n k ( t ) (cid:18) − i √ (cid:19) n H n i (cid:16) z a † k + z α ( ψ ) k ( t ) (cid:17) √ z z (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) , (2.47) where H n ( x ) are the Hermite polynomials, now expressed in terms of the creation operator a † k for any given mode k . The coherent states (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) , on the other hand, may beexpressed as linear combinations of the eigenstates exp (cid:104) − i (cid:0) n + (cid:1) ω ( ψ ) k t (cid:105) | n ; k , ψ k (cid:105) , whichimmediately ties up (2.47) to (2.35). This means the coefficients C ( ψ ) n k ( t ) of (2.47) shouldbe related to the coefficients c ( ψ ) n ( k , t ) of (2.35). The relation is not too hard to find, andmay be expressed as: c ( ψ ) m ( k , t ) = ∞ (cid:88) n =0 ( − i ) n C ( ψ ) n k ( t ) H n i (cid:115) ω ( ψ ) k z z z f k − z b ∂∂f ∗ k + z α ( ψ ) k ( t ) (cid:113) ω ( ψ ) k exp − (cid:12)(cid:12)(cid:12) α ( ψ ) k ( t ) (cid:12)(cid:12)(cid:12) (cid:16) α ( ψ ) k (0) (cid:17) m √ n m ! , (2.48) Recall that the Agarwal-Tara state [36] is for the limit where ( z , z ) = (0 ,
1) and C ( ψ ) n k ( t ) = δ nm with m ∈ Z . Here we will explore a more generalized version of this by keeping both z i non-zero and switchingon time-dependent coefficients. – 31 –here b is the same parameter that appears in the vacuum wave-function (2.30) and wehave expressed the Hermite polynomial of operators in terms of Schr¨odinger representa-tion so that it can directly act on the wave-functions in the configuration space. In thissense (2.48) is more like an operator relation where the LHS should be thought of as anidentity operator modulated by the constant factor of c ( ψ ) m ( k , t ) whereas the RHS is a sumof operators in Schr¨odinger formalism. One could in principle work out an operator freerelation between the coefficients c ( ψ ) m ( k , t ) and C ( ψ ) n k ( t ) by acting the creation operator a † k inside the Hermite polynomial iteratively from (2.47), but since this will lead to no newphysics beyond the fact that the coefficients are connected, we will refrain from indulgingin a more convoluted exercise here. Instead, from the fact that the two states (cid:12)(cid:12)(cid:12) Ψ ( αg ) k (cid:69) from(2.47) and (cid:12)(cid:12)(cid:12) Ψ ( ψ ) k (cid:69) from (2.35) are related, we can then propose that the state (cid:12)(cid:12)(cid:12) Ψ ( c c ) k (cid:69) from (2.43) may be directly related to the coherent state (cid:12)(cid:12)(cid:12) Ψ ( α ) k (cid:69) via the following relation: (cid:12)(cid:12)(cid:12) Ψ ( c c ) k ( t ) (cid:69) = (cid:104) c + c G ( ψ ) ( a k + a † k ; t ) (cid:105) (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69) , (2.49)where the operator G ( ψ ) ( a k + a † k ; t ) is defined in (2.46). The relation (2.49) is valid for alltime t , and we get pure coherent state for vanishing c . The relation (2.49) also leads usto conclude the following:Four-dimensional de Sitter space is a Glauber-Sudarshan state [26, 27] in string theory,or alternatively, a coherent state in string theory. Similarly fluctuations over a de Sitterspace appear from a generalized
Agarwal-Tara state [36], or alternatively, from a gen-eralized graviton-added coherent state. Both these descriptions are over supersymmetricMinkowski backgrounds, or more generically, over supersymmetric solitonic backgrounds.In the rest of the paper we will make the above statement more precise and concreteby answering many issues that may arise in an actual realization of de Sitter space as acoherent state.The realization (2.49) tells us that the fluctuations over a de Sitter space in our analysiscan be inferred by adding extra gravitons to our coherent state. Recall that the coherentstate is already a condensation of gravitons, so the natural question is to ask about thenumber of gravitons in a state like (2.49). Such an analysis should shed some light not onlyon the entropy of the de Sitter space itself but also on how the entropy changes by addingfluctuations over the de Sitter space. First however we should determine the total numberof gravitons packed in the coherent states for all the allowed accessible modes k . Thewave-function for such a state may be represented by the following integral representation:Ψ ( α ) ( f, t ) = (cid:68) f (cid:12)(cid:12)(cid:12) Ψ ( α ) ( t ) (cid:69) ≡ exp (cid:18)(cid:90) + ∞−∞ d k log (cid:68) f k (cid:12)(cid:12)(cid:12) Ψ ( α ) k ( t ) (cid:69)(cid:19) , (2.50)where the ket | f (cid:105) denotes the coordinates in the configuration space for all the modes k . We have also used the simplifying notation as in (2.45) which, although useful to Interestingly, for non-vanishing c but vanishing z in (2.46), we get back the coherent state. – 32 –void clutter, loses the information about the fact that it is only the ψ k ( x , y, z ) part ofthe whole system. This means the above wave-function is still not the full wave-functionin the configuration space that reproduces the M-theory space-time metric configuration(2.4) as the most probable outcome. Nevertheless it is a useful guide for what is about tofollow. For example, the number N ( ψ ) of gravitons in such a state is then the standardexpectation value of the number operator over the state (2.50). Since we know the precisewave-functions for every mode k from (2.13), the number of gravitons becomes: N ( ψ ) ≡ (cid:90) + ∞−∞ d k (cid:12)(cid:12)(cid:12) α ( ψ ) k (0) (cid:12)(cid:12)(cid:12) , (2.51)which, as we warned before, is not the full answer yet. It only tells us about the number ofgravitons with space-time wave-functions as in (2.33). What about the number of gravitonsin a state like (2.47)? Can we pack arbitrary number of gravitons in such a state? This iswhere the issue of back-reaction comes in.A necessary requirement for the GACS to be identified as fluctuations over a de Sitterspace-time would be that the back-reaction corresponding to this state is under control. Inusual perturbation theory over de Sitter, the standard back-reaction constraint for such asystem would be to imply that the energy in the fluctuation fields are much smaller thanthe energy density of the background. In our case, the analogous relation would imply thatthe energy density of the fluctuations in the GACS state has to be significantly smallercompared to the energy density of our Glauber-Sudarshan state. The subtlety for ourconstruction lies in the fact that both these states are constructed over a solitonic vacuumand one cannot use a simple expression, such as M p H , to characterize the Hubble scale ofthe background .Having said this, it is easy to see that the energy in the wave-function (2.13) can becalculated analogously to the way the number of gravitons were calculated in (2.51), andis given by: E ( ψ ) ≡ (cid:90) ∞−∞ d k ω ( ψ ) k (cid:12)(cid:12)(cid:12) α ( ψ ) k (0) (cid:12)(cid:12)(cid:12) , (2.52)keeping in mind, as usual, that this only corresponds to gravitons with space-time wave-functions as in (2.33). In a similar vein, one can calculate the energy of the gravitonspacked in the GACS state (2.36) as: E ( α g ) ≡ (cid:88) n (cid:90) ∞−∞ d k ω ( ψ ) k n (cid:12)(cid:12)(cid:12) c ( ψ ) n ( k , (cid:12)(cid:12)(cid:12) . (2.53)At first sight, it might seem odd that there is no α ( ψ ) k dependence of this expression sincewe have stressed that fluctuations over de Sitter space as a GACS. However, keep in mindthat while expressing the state (2.36) as a generalized Agarwal-Tara state in (2.49), wehave expressed the coefficients c ( ψ ) n in terms of the coefficients C ( ψ ) n k (see (2.48)), we had to Although, note that, we expect such an effective description to emerge from our Glauber-Sudarshanwave-function just as a cosmological constant is emergent in this case. – 33 –nclude several terms which depend on α ( ψ ) k . Now that we are assured of the consistencyof our treatment, let us finally recall that our generic state from (2.43), or in (2.49), comeswith (constant) pre-factors c and c .This is all good, and points towards the consistency of our treatment, both for the deSitter space viewed as a coherent state and fluctuations over the de Sitter space viewed asa GACS. Therefore, given these results, it is easy to express our backreaction constraintas: c E ( ψ ) (cid:29) c E ( α g ) . (2.54)It is important for our generic state defined in (2.49) to always satisfy the above conditionfor it to be able to describe small fluctuations over de Sitter space-time. The limitingcondition of c → , c → all allowed momenta, implying an access to arbitrarily short distances. Such an analysishas to be reconsidered in the light of the Wilsonian effective action, which allows us toaccess momenta | k | ≤ M p . The modes lying between M p ≤ | k | ≤ Λ UV , where Λ UV is theshort distance cut-off, are integrated out resulting in a non-trivial effective action. Doesour de Sitter space, resulting from the coherent state construction, survive the tower ofquantum corrections coming from integrating out momentum shell from the cut-off Λ UV to M p ? This question clearly cannot be answered from what we did so far: while certaininteractions were entertained in the construction of the solitonic vacuum and from therethe coherent states, our analysis no way mixes the k and k (cid:48) modes in any way. In fact notonly the modes don’t mix, the various sectors represented by the four set of wave-functionsin (2.7) do not mix either. However this is not the only short-coming: the worse is yet tocome. There are also modes coming from the G-fluxes, that will have their own sectorsrepresented by similar spatial wave-functions. These modes should mix amongst eachother, and they should also mix with the modes of the gravitational sector that we mademeticulous efforts to construct in the previous sections. In addition to that there are alsohigher order perturbative and non-perturbative, as well as local and non-local, quantumcorrections (including topological ones!). How do we know, when we let everything mixamongst each other, the de Sitter state would survive in the final theory?The above question looks almost like an impossible question to answer, but if wecarefully analyze the situation, this may not be that difficult. In the following section wewill start by discussing one possible approach to address this question. A way to address question like this is to start by laying out all the contents of our inter-acting configuration space. Needless to say, from eleven-dimensional point of view, these– 34 –nteractions are going to be highly non-trivial. First, however there is some light at theend of the tunnel: the Wilsonian analysis can be performed because the modes, at leastwhat we argued from (2.7), do not have time-dependent frequencies although their spatialwave-functions could be non-trivial implying, in turn, that there are no trans-Planckianissues plaguing our analysis. Secondly, there is a possibility that mixing of the modes fromeach sector ( i.e. from the gravitational as well as the G-fluxes) will allow us to create newsectors with their own mixed spatial wave-functions, and with new creation and annihila-tion operators, on which we can have our coherent states. The vacuum of the mixed sectorwill be non-trivial, which is nothing but the interacting vacuum generated from: | Ω( t ) (cid:105) ∝ lim T →∞ (1 − i(cid:15) ) exp (cid:18) − i (cid:90) t − T d x H int (cid:19) | (cid:105) , (2.55)where H int is the interacting Hamiltonian in M-theory that we will specify soon. In fact H int contains all information about the local and non-local, that include the perturbative,non-perturbative and topological, quantum corrections. The state that we are looking for,in light of what we discussed earlier, and in the fully interacting theory, may be expressedas: Ψ ( α )Ω ( f, t ) ≡ (cid:10) f (cid:12)(cid:12) D ( α ( t )) (cid:12)(cid:12) Ω( t ) (cid:11) , (2.56)where the ket | f (cid:105) , as before, is the coordinate of the interacting configuration space, | Ω( t ) (cid:105) is the same interacting vacuum as in (2.55), and D ( α ( t )) is the displacement operator that shifts the interacting vacuum in the configuration space. The question then is: does thiscreate a coherent state in the interacting theory?Even before we start answering this question, the very meaning of a displacementoperator in an interacting theory is not clear. From the mode-by-mode analysis that wedid above, an interacting theory will not only be highly anharmonic, to say the least,but will also have interactions between the modes themselves. Such interaction wouldtypically take us away from the simple-harmonic-oscillator regime, but if the interactionshave perturbative expansions then we can at least formally write: D ( α ( t )) (cid:12)(cid:12) Ω( t ) (cid:11) ∝ D ( α ( t )) | (cid:105) + ∞ (cid:88) n =1 ( − i ) n n ! D ( α ( t )) (cid:90) t − T dt ....dt n T (cid:40) n (cid:89) i =1 (cid:90) d x i H int ( t i , x i , y i , z i ) (cid:41) | (cid:105) , (2.57) where in the second term we still have to allow T → ∞ (1 − i(cid:15) ) to avoid other interactingstates to emerge in the sum; and T denotes time-ordering. Here | (cid:105) is the solitonic vacuumwhose wave-function may be described as:Ψ ( f ) ≡ (cid:104) f | (cid:105) = exp (cid:18)(cid:90) + ∞−∞ d k log Ψ ( f k ) (cid:19) , (2.58)with | f (cid:105) denoting the coordinates in the configuration space with a wave-function Ψ ( f k )for every mode k . This wave-function will have further finer sub-divisions if we want to Even further if we want to incorporate all the spatial wave-functions describing the components ofG-fluxes. We will avoid these complications for the time being and deal with them a bit later. – 35 –escribe all the spatial wave-functions in (2.7) for any given mode k . Once we know theground state wave-function, the constants of proportionalities in (2.55) and (2.57) are bothrelated to the overlap function (cid:104) Ω( t ) | (cid:105) .The above identification (2.57) would still make no sense unless we identify the dis-placement operator D ( α ( t )) for the interacting theory. In our earlier analysis with coherentstates, the displacement operator is defined by exponentiating the creation and the anni-hilation operators. For a highly interacting theory, there is no simple description of thecreation and the annihilation operators, but we can define two operators that take thefollowing form: a eff ( k , t ) = a k + (cid:88) l,n,m (cid:90) d k .....d k n d k (cid:48) .....d k (cid:48) m c lnm f l k (cid:16) a k .....a k n a † k (cid:48) .....a † k (cid:48) m ; t (cid:17) a † eff ( k , t ) = a † k + (cid:88) l,n,m (cid:90) d k .....d k n d k (cid:48) .....d k (cid:48) m c ∗ lnm f † l k (cid:16) a k .....a k n a † k (cid:48) .....a † k (cid:48) m ; t (cid:17) , (2.59)such that a eff ( k , t ) annihilates the interacting vacuum | Ω( t ) (cid:105) . Here ( a k , a † k ) are as definedin (2.32); and the conjugate transpose action on f l k ( ... ) acts in a standard way by complexconjugating the coefficients and converting a k → a † k with due considerations to the orderingof the operators. The operator definition in (2.59) makes sense if all the dimensionlesscoefficients multiplying the operator products in the definition of f l k are smaller than c lnm and c ∗ lnm ; and additionally Re ( c lnm ) << Im ( c lnm ) <<
1. If this be the case , thenthe commutator brackets: (cid:104) a eff ( k , t ) , a † eff ( k (cid:48) , t ) (cid:105) = δ ( k − k (cid:48) ) + O ( c lnm ) (cid:2) a eff ( k , t ) , a eff ( k (cid:48) , t ) (cid:3) = O (cid:0) c lnm (cid:1) = (cid:104) a † eff ( k , t ) , a † eff ( k (cid:48) , t ) (cid:105) , (2.60)would tell us how far are we from a simple-harmonic-oscillator description in the configu-ration space. Note three things: one, the first commutator in (2.60) has O ( c lnm ) correctionterm whereas the second commutator has O ( c lnm ) correction term; two, due to the O ( c lnm )correction term, a † eff ( k , t ) cannot be defined as a standard creation operator like a † k ; andthree, the appearance of t in the definitions of the operators in (2.59). The former implies This will still not fix the form of a eff unambiguously unless more conditions are specified. For the timebeing it will suffice to assume that at least a particular a eff exists in the theory that mixes all the harmoniccreation and annihilation operators for each modes k in the simplest possible way. More elaborations onthis will be dealt in section 3.3. . The coefficients c lnm are related to the coupling constants of the theory and therefore one would expectnon-perturbative corrections like c lnm to appear too ( n, l and m are not Lorentz indices!). In string theory, c lnm , or any other coefficients, can only be proportional to g s , the string coupling (which we shall specifylater). As such one expects non-perturbative series in g s to appear. However these inverse g s factorscan be resummed as a resurgent trans series to exp (cid:16) − g / s (cid:17) , or to exp (cid:16) − c lnm (cid:17) here. They becomearbitrarily smaller than any polynomial powers of g s or c lnm in the limit g s << Re ( c lnm ) << Im ( c lnm ) << – 36 –tronger suppression of the second commutator whereas the latter means the commutationrelations (2.60) continue for the range of time that keeps all the dimensionless coefficientsentering in the definition (2.59) under perturbative control implying, in turn, that the twooperators in the second set of commutators of (2.59) remain orthogonal when k (cid:54) = k (cid:48) up-to O (cid:0) c lnm (cid:1) . Therefore using (2.60) we can give the following operator definition of D ( α ( t )): D ( α ( t )) ≡ exp (cid:18) − | α | (cid:19) exp (cid:16) αa † eff (cid:17) exp ( − α ∗ a eff ) (2.61)= exp (cid:34) αa † eff − α ∗ a eff − | α | + ∞ (cid:88) n =1 n ( n + 1) (cid:90) dh (cid:16) I − exp( L αa † eff )exp ( h L − α ∗ a eff ) (cid:17) n α ∗ a eff (cid:35) which would be similar in spirit of the definition of a displacement operator that was used togenerate the coherent states earlier . We will call the shifted interacting vacuum D ( σ ) | Ω (cid:105) as the generalized Glauber-Sudarshan state to distinguish it from the original Glauber-Sudarshan state created out of the shifted harmonic vacuum D ( σ ) | (cid:105) . The second equalityin (2.61) comes from using the generic form of the Baker-Campbell-Hausdorff relation. Theother terms may be defined as follows. I is the identity operator (in the relevant basis),and we have used the following definitions of the products of operators and parameters in(2.61): αa † eff = (cid:90) d k e µν ( k , s ) α ( ψ ) µν ( k , t ) a † eff ( k , t ) , | α | = (cid:90) d k α ( ψ ) µν ( k , t ) α ∗ ( ψ ) µν ( k , t ) (2.62) L αa † eff ( α ∗ a eff ) = (cid:90) d k d k (cid:48) e µν ( k , s ) e ∗ σρ ( k (cid:48) , s ) α ( ψ ) µν ( k , t ) α ∗ ( ψ ) ρσ ( k (cid:48) , t ) (cid:104) a † eff ( k , t ) , a eff ( k (cid:48) , t ) (cid:105) , where, as before, the repeated indices are not summed over as we are only dealing withthe wave-functions (both spatial and in the configuration space) associated with the metricmode g µν ( x, y, z ). Note that the operator action of L αa † eff on α ∗ a eff from (2.62), whencombined with (2.60), generates one term that exactly cancels the − | α | piece in (2.61).This is as it should be, but now we see that more involved operator products also emergein addition to the expected answers. However one would also like to compare the difference operator: Q ( ψ )1 ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp (cid:16) αa † eff (cid:17) − (cid:89) k exp (cid:16) e µν ( k , s ) α ( ψ ) µν ( k , t ) a † eff ( k , t ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.63)where for simplicity we have taken discrete momenta k to give meaning to the second termabove. We can similarly define Q ( ψ )2 by the following replacement: a † eff → a eff , α → α ∗ and α ( ψ ) µν → α ∗ ( ψ ) µν appropriately in (2.63). The operator Q ( ψ )1 is identically zero for free theory Another definition of the displacement operator may be given by taking only the first two terms from(2.61), namely D ( α ( t )) = exp (cid:16) αa † eff − α ∗ a eff (cid:17) . This definition, although closer in spirit to the definition ofdisplacement operator in free theory, does not reproduce the simple product relation in the first equalityof (2.61). In other words, if we want to express this definition of the displacement operator also as e A e B where ( A , B ) are two operators, they cannot be some simple combinations of a eff and a † eff . Clearly eitherdefinitions would work in the limit c lnm →
0, but since we wish to explore dynamics governed by c lnm < – 37 –nd can be made arbitrarily small when c lnm →
0, and therefore serves as a signature ofhow interacting the theory is. One may similarly define the conjugate difference operator | Q ( ψ )2 − Q † ( ψ )1 | which do not vanish either. Either of these serves as a good focal point tostudy interactions, but more importantly they signify how efficiently one may study thecoherent states in an interacting theory using the operator definition (2.61).With these, we are almost ready to write down the wave-function in the configurationspace for the interacting vacuum | Ω( t ) (cid:105) when it is displaced by an amount α ( ψ ) µν ( k , t )for any mode k . However since the modes mix non-trivially, the wave-function should beexpressed, not in terms of individual modes k , but in terms of the field itself. The field inquestion is the space-time metric component g µν ( x , y, z ) at any given instant of time. Inthe limit where a eff ( k , → a k and a † eff ( k , → a † , i.e. when c lnm = 0 (see footnote 8) thedisplacement operator is the free-field one and will be denoted by D ( α ( t )). This meanswe can define an operator : δ D ( α ( t )) = D ( α ( t )) − D ( α ( t )) , (2.64)which is by construction perturbatively controlled by the coefficients c lnm , with D ( α ( t )) asin (2.61). It is also clear that there is a free field displacement operator for any mode k ,and they generically commute amongst each other. Therefore putting everything together,the wave-function for the shifted interacting vacuum can be expressed as: Ψ ( α )Ω ( g µν , t ) = exp (cid:20)(cid:90) + ∞−∞ d k log (cid:16) Ψ ( α ) k ( (cid:101) g µν ( k ) , t ) (cid:17)(cid:21) + (cid:90) D g (cid:48) µν (cid:104) g µν (cid:12)(cid:12) δ D ( α ( t )) (cid:12)(cid:12) g (cid:48) µν (cid:105) Ψ ( g (cid:48) µν ) (2.65)+ (cid:88) n ( − i ) n n ! (cid:90) D g (cid:48) µν D ˆ g µν (cid:104) D ( α ( t )) (cid:105) (cid:90) t − T dt .....dt n (cid:104) ˆ g µν (cid:12)(cid:12) T (cid:40) n (cid:89) i =1 (cid:90) d x i H int ( t i , x i , y i , z i ) (cid:41) (cid:12)(cid:12) g (cid:48) µν (cid:105) Ψ (cid:0) g (cid:48) µν (cid:1) , Here as a trial example we are displacing by an amount α ( ψ ) µν ( k , t ) from (2.17). As we shall see soon,such a choice reproduces correct answers up to O (cid:16) g | a | s M bp (cid:17) corrections. The actual choice of shift that doesnot allow extra corrections terms is more subtle and we will have to wait till section 3.3 to get more exactresults. For the time being this will suffice. Expressing the displacement operator (2.61) as a linear combination of the free-field displacementoperator D ( α ) and perturbatively controlled corrections δ D ( α ), has an additional advantage of elucidatingsome of the properties expected of this operator. They are listed as follows. D − ( α ) = D − ( α ) − D − ( α ) δ D ( α ) D − ( α ) + .... D † ( α ) = D † ( α ) + ( δ D ) † ( α ) , D ( − α ) = D ( − α ) + δ D ( − α ) , which would tell us that while the equalities D † ( α ) = D − ( α ) = D ( − α ) are exact, similar equalities for D ( α ) are only true to O ( δ D ( α )), implying that D ( α ) is only approximately unitary. In a similar vein onemay easily see that: D † ( α ) a eff D ( α ) = (cid:16) D † ( α ) + ( δ D ) † ( α ) (cid:17) ( a + O ( c lnm )) ( D ( α ) + δ D ( α ))= a + α + O ( δ D ( α )) + O (cid:16) δ D † ( α ) (cid:17) + O ( c lnm , δ D ( α )) + O (cid:0) | δ D ( α ) | (cid:1) , where the first two terms in the second equality would relate to the expected identity if one replaces D ( α )by D ( α ). Taking the conjugate of the operator relation above would provide yet another identity. Clearlysince the RHS of the above equation do not amount to a eff + α , or, from its conjugate identity, a † eff + α ∗ ,the state created by the action of D ( α ) on the interacting vacuum | Ω (cid:105) will not be a coherent state, at leastnot exactly. This will also become clear from the wave-function (2.65) discussed below. – 38 –here (cid:104) D ( α ( t )) (cid:105) ≡ (cid:104) g µν (cid:12)(cid:12) D ( α ( t )) (cid:12)(cid:12) ˆ g µν (cid:105) , which is similar to how we defined the expectationvalue of δ D ( α ) above. The other parameters appearing above are the wave-functions thatare defined as follows: Ψ (cid:0) g (cid:48) µν (cid:1) is the vacuum wave-function in the configuration spacedefined over the free vacuum exactly as in (2.58); and Ψ ( α ) k ( (cid:101) g µν ( k ) , t ) is the coherent statewave-function that we derived earlier in (2.13). Here, as promised above, we have definedthe wave-functions for all modes, but restricted ourselves to only the space-time compo-nent g µν . There exists a more complete wave-function for all modes and all components,including the ones from the G-fluxes, but we will not discuss this right now. In fact thefull wave-function is not necessary as (2.65) is enough to elucidate the main point of ouranalysis: the dominant part of the shifted vacuum wave-function for the interacting theoryis indeed given by the Glauber-Sudarshan wave-function for the harmonic theory. Theremaining two terms of (2.65) are the perturbative corrections controlled by c lnm from(2.61).The third term that appears in (2.65) is interesting by itself. It involves the interactingHamiltonian H int from M-theory and is thus a much more complicated object. The actionof this interacting Hamiltonian, acting on the field states | g µν (cid:105) , is an integrated action,namely, that we integrate from − T = −∞ (in a slightly imaginary direction) to the presenttime t (or more appropriately √ Λ t to make this dimensionless). There are clearly threepossible outcomes of such an integrated action:(1) The integrated action of the Hamiltonian H int over the range of time from − T = −∞ till the present epoch √ Λ t , along-with the effects from (2.64), exactly cancels out so thatthe system continues to evolve classically over an indefinite period of time.(2) The combined action of the interacting Hamiltonian and the operator (2.64), over theintegrated period of time, allows the classical feature to persist exactly for a certain intervalof time beyond which the system becomes truly quantum. The point of time at which suchclassical to quantum transition happens should be related to the quantum break time ofthe system .(3) The combined action of the interacting Hamiltonian and the operator (2.64), do con-tribute non-zero values over the integrated period of time, but never enough to changesubstantially the behavior of the mode-by-mode wave-function (2.13). As such the systemcontinues to allow classical configurations, like (2.4) with perturbative corrections, eitheras most probable outcomes or as expectation values.It is clear that we can safely disregard option 1, as it is highly unlikely that quantumcorrections could cancel exactly to allow for the quantum system to evolve classically and It is interesting to compare the quantum break time as proposed in [29] and the one that we want toconsider in (2.80). Our choice of the temporal domain is motivated by the behavior of the string coupling,or more appropriately g s √ h ( y ) , till it hits strong coupling because beyond that we have no control on thedynamics. The behavior of the Glauber-Sudarshan state also becomes out of control once strong couplingsets in. This noticeable different approach of dealing with the quantum break time from [29] has it’s rootsin the actual string dynamics underlying the system. – 39 –ver an indefinite period of time. The choice then is between options 2 and 3. In thefollowing we will discuss which of the two options would favor our system. We will start by revisiting the computation of the expectation value in (2.16), but now usingthe path-integral formalism so that we can generalize this to extract relevant informationfrom the displaced interacting vacuum (2.57).We will start with a simple example from free massive scalar field theory in 3 + 1dimensions with a mass term given by m . Our aim would be to reproduce the expectationvalue of the scalar field ϕ on a coherent state | α (cid:105) , using path integral formalism. Thevacuum of this theory is simple, its given by ϕ = 0, which makes the analysis even simpler.In fact it also makes ψ k ( x ) = exp ( − i k · x ). We could of course generalize this to allow morenon-trivial solitonic solution, and go to the harmonic oscillator regime where the spatialwave-functions are given by a non-trivial function ψ k ( x ), but will avoid these complicationsfor the time being. However since we want to use path integrals, we will resort to a moreoff-shell description of the fields as the following Fourier expansion: ϕ ( x ) = (cid:90) d k ϕ ( k , k ) exp ( ik · x ) , (2.66)where we have used the mostly minus signature for this example, and k ≡ ( k , k ). Thereis no relation between k and k right now, so the fields are off-shell. We will also need toexpress the creation and the annihilation operators (2.32) in terms of the Fourier modes(2.66) without involving any derivatives because the path integral formalism prefers fieldsinstead of operators. In fact this is not hard and the answer is to replace these operatorsby: a k = (cid:114) ω k (cid:18) ˆ ϕ k + i ˆ π k ω k (cid:19) → ( ω k + k ) ϕ k a † k = (cid:114) ω k (cid:18) ˆ ϕ ∗ k − i ˆ π ∗ k ω k (cid:19) → ( ω k + k ) ϕ ∗ k , (2.67)where, the hatted quantities are the operators ; and we have used a slightly differentdefinition from (2.32) because of the change of signature, and ϕ k ≡ ϕ ( k , k ). The conjugate Note the absence of √ ω k in the second equalities of (2.67), although the first equalities demand it.This is because the standard mode expansion in field theory appears with a measure d k √ ω k , and thus thecreation and the annihilation operators are required to have this as prefactors (see (2.32)). Once we goto the field description we allow off-shell quantities; however this is a bit puzzling from (2.29) which isexpressed in terms of on-shell quantities in the Heisenberg formalism (the integral is also d d k and not d d +1 k ). A simple way out is to define the measure as d k ω k ≡ (cid:82) k d k δ ( k − m ) so that it may be expressedas a four-dimensional integral. One can then claim that the operator form, i.e. using (2.32), is a specialcase where we take a four-dimensional integral with fields and put it on-shell when we replace the fieldsby the operators (2.32). This introduces a pre-factor of ω k in the second equalities of (2.67) which cancelagainst the inverse ω k factors in the definition of the conjugate momentum. The extra factor of , whichappears from (2.29), cancels out also leaving us with the two rightmost quantities in (2.67) without anyother numerical factors. It also matches up with (2.13) as well as with (2.68). As we shall see below, thishelps us to get the appropriate residue at the pole k = ω k . – 40 –omentum π k ≡ − ik ϕ k , and note the appearance of ω k + k , ω k being the usual √ k + m , which is the sign that this is an off-shell definition. In fact these definitionshelp us to express : exp (cid:16) αa † (cid:17) → exp (cid:20)(cid:90) + ∞−∞ d k ( ω k + k ) (cid:101) α k ϕ ∗ k (cid:21) , (2.68)and similar description for the conjugate operator exp ( α ∗ a ). Such a description is usefulbecause we can define the coherent states as an action of ˆ D ( α ) over the free vacuum inthe following way: ˆ D ( α ) | (cid:105) ≡ exp (cid:18) αa † − | α | (cid:19) | (cid:105) , (2.69)with | (cid:105) being the free vacuum, with ˆ D ( α ) differing from D ( α ) by not being unitary. Thiswill necessitate a division by (cid:104) α | α (cid:105) in the path-integrals to keep the normalization straight.Combining everything together, the expectation of the scalar field on the coherent statemay be expressed as: (cid:104) ϕ ( x ) (cid:105) α = (cid:104) α | ϕ ( x ) | α (cid:105)(cid:104) α | α (cid:105) = (cid:82) D ϕ e iS ˆ D † ( α ) ϕ ( x ) ˆ D ( α ) (cid:82) D ϕ e iS ˆ D † ( α ) ˆ D ( α ) , (2.70)where S is the action for the free scalar field and here it may be written as an integral ofthe form S ≡ (cid:82) d k ( k − m ) | ϕ k | . The path integral (2.70) involve various nested integralsand therefore to perform the integrals efficiently it will be advisable to use discrete sum.This is all very standard, so we simply show one step of how to express the numerator ofthe path integral (the second equality in (2.70)) in the following way : (cid:90) D ϕ e iS ˆ D † ( α ) ϕ ( x ) ˆ D ( α ) = (cid:32)(cid:89) k (cid:90) d ( Re ϕ k ) d ( Im ϕ k ) (cid:33) exp (cid:34) iV (cid:88) k (cid:0) k − m (cid:1) | ϕ k | (cid:35) (2.71) × exp (cid:34) V (cid:88) k (cid:48) (cid:0) ω k (cid:48) + k (cid:48) (cid:1) ( Re (cid:101) α k (cid:48) Re ϕ k (cid:48) + Im (cid:101) α k (cid:48) Im ϕ k (cid:48) ) (cid:35) × V (cid:88) k (cid:48)(cid:48) exp (cid:0) ik (cid:48)(cid:48) · x (cid:1) ( Re ϕ k (cid:48)(cid:48) + i Im ϕ k (cid:48)(cid:48) ) exp (cid:32) − V (cid:88) k (cid:48) α k (cid:48) α ∗ k (cid:48) (cid:33) , where V is the volume of the four-dimensional space (which is finite and that makes allmomenta discrete so that the sum could be performed). Even without introducing any Here regarded as a field and not as an operator. The expression inside the exponential factor in (2.68) appears to break Lorentz invariance, becausewe chose to insert the factor ω k + k from the definition (2.67). In fact Lorentz invariance is actually notbroken because the quantity that appears on the exponential is not α k , rather (cid:101) α k . They are related by: (cid:101) α k ≡ ( k − ω k ) α ( k ) + α k k + ω k where α ( k ) is an arbitrary function of k . For the cases that we would be interested in, α ( k ) = 0, so theterm in the exponential will be simply α k ϕ ∗ k , which in turn is a Lorentz invariant quantity. We will analyze the path integral using Lorentzian signature although, from Wilsonian sense Euclideansignature serves better. We will resort to this a bit later when we actually discuss the Wilsonian effectiveaction. – 41 –implifications, we see that if the momenta ( k, k (cid:48) , k (cid:48)(cid:48) ) are all different the integral (or moreappropriately the sum) vanishes . Thus the above integral can only give non-zero values ifthe momenta at every stage of the sum are related to each other. This is a crucial pointand deserves some explanation. In the standard path integral, integrating a term linear in ϕ k would vanish because the gaussian functions from the action are all even functions.Here however there is an additional exponential piece which is expressed by linear powersof ϕ k . As such this would shift the center of the gaussian giving a non-zero value for theintegral. In a similar vein the denominator of the path integral in (2.70) will be exactlysimilar to (2.71) without the last term. Now to determine the precise value of the integral(2.71) we can make a simplifying assumption that (cid:101) α k ’s are real. The quadratic pieces in ϕ k ’s then produce the necessary poles at k = ω k , rendering: (cid:104) ϕ ( x ) (cid:105) α = (cid:90) d k ω k Re α k exp ( iω k t − i k · x ) , (2.72)where the dk integral takes care of the poles by inserting appropriate residues, with theoff-shell part coming from an expression similar to (2.17). The result is close to what wehad in (2.16) once we replace the graviton field g µν by the scalar field and go to four-dimensions. The graviton description is over a solitonic background and thus we see thatexp ( i k · x ) in (2.72) is to be replaced by ψ k ( x ) (the two results (2.72) and (2.16) appearto be complex conjugates of each other, once we replace ψ k ( x ) by ψ − k ( x ), because of thechange of signature used here, as defined earlier).This roundabout way of getting the simple result (2.72) (or (2.16)) is not without itsmerit. It prepares us to tackle a much more involved problem associated to the displacedinteracting vacuum | Ω( t ) (cid:105) . However before moving ahead it will be instructive to resolveone puzzle associated to the path integral computation that we presented above. Thepuzzle involve the usage of ˆ D ( α ) instead of D ( α ) in (2.70). If we had used D ( α ), theintegral (2.70) or (2.71) would have vanished because of the unitary nature of D ( α ) (it isno longer an operator in the Feynman formalism as seen from (2.68)). The reason for thisapparent discrepancy lies on the following redundancy of the free vacuum | (cid:105) : f ( a ) | (cid:105) ≡ (cid:32) ∞ (cid:88) n =1 c n a n (cid:33) | (cid:105) = | (cid:105) , (2.73)for n ∈ Z + but c n any positive or negative number, with a being the annihilation operator.This means that there is always an ambiguity when we choose D ( α ): both D ( α ) and D ( α ) G ( f ( a )), with G ( f ) being any functional of f ( a ), would create the same coherentstate implying, we can always use G ( f ) to eliminate the a dependence of D ( α ) and insteaduse the definition (2.69), which is non-unitary. If we make the same change in (2.64), then D ( α ( t )) in (2.61) may be re-expressed as D ( α ) = ˆ D ( α )+ δ D ( α ), which is again non-unitary.Our aim now is to figure out which of the two options, options (2) or (3), in the previoussub-section, would qualify our system. For that we want to compare the expectation valueof the metric operator on the coherent state | α (cid:105) . In other words we want to determine (cid:104) g µν (cid:105) α with the assumption that: D ( α ( t )) | Ω( t ) (cid:105) ≡ | α (cid:105) , (2.74)– 42 –nd compare it with (2.70). Our aim would be to see how much the expectation value differsfrom the expected answer (2.16). Any deviation from (2.16) would imply the deviation ofthe configuration space wave-function from the Glauber-Sudarshan wave-function due tothe interaction Hamiltonian H int . As such this might help us to quantify the contributionsfrom the interaction Hamiltonian.We will start by assigning an off-shell description of the metric component g µν as in(2.18). This is similar to what we did for the scalar field case in (2.66). One immediateuse of such a description is that it helps us to express the operators by fields, in the samevein as in (2.68). In our case we expect :exp (cid:16) αa † eff (cid:17) → exp (cid:20)(cid:90) + ∞−∞ d k α ( ψ ) µν ( k , k ) (cid:101) g ∗ µν ( k , k ) + ...... (cid:21) , (2.75)where no sum over the repeated indices are implied here and the dotted terms are higherorders in α ( ψ ) µν ( k , k ) and (cid:101) g µν ( k , k ) that could mix the momenta as well as the spatial indicesas expected from (2.59). One should also compare (2.75) to (2.62) where e µν ( k , s ) a † ( k , t )in (2.62) goes in the definition of the operator (cid:101) g µν ( k , t, s ) which is then converted to theoff-shell field (cid:101) g µν ( k , k ) that appears in (2.75).Our next couple of steps will be similar to what we had for the simple scalar field theory,namely, we define the operator D ( α ) as in (2.69) with appropriate replacements from (2.75)and (2.62). This is of course motivated from the fact that a eff ( k , t ) | Ω( t ) (cid:105) = 0 and thereforethe interacting vacuum has similar redundancy as in (2.73). This reproduces : D ( α ) = exp (cid:20)(cid:90) + ∞−∞ d k α ( ψ ) µν ( k , k ) (cid:101) g ∗ µν ( k , k ) − (cid:90) + ∞−∞ d k α ( ψ ) µν ( k , k ) α ∗ ( ψ ) µν ( k , k ) + .... (cid:21) , (2.76) where the repeated indices are not summed over, and the dotted terms carry over from(2.75). The eleven-dimensional integral tells us that the quantities appearing in (2.76) ismaximally off-shell. Putting everything together, the expectation value for metric compo-nent over the coherent state becomes: (cid:104) g µν ( x, y, z ) (cid:105) α = (cid:82) [ D g µν ] e iS D † ( α ) g µν ( x, y, z ) D ( α ) (cid:82) [ D g µν ] e iS D † ( α ) D ( α ) , (2.77)where S is now the full interacting lagrangian of M-theory (which we will specify a bitlater), and the first term from evaluating the path integral is the solitonic background See footnote 31 and 26. Its interesting to note that the form of the displacement operator D ( α ), as given by the first term in(2.76), is somewhat similar to exponentiating the vertex operator V to create gravitons in string perturbationtheory, namely:exp ( V ) = exp (cid:18) πα (cid:48) (cid:90) d σ √ h h ab ∂ a X µ ∂ b X ν g µν (cid:19) ≡ exp (cid:18)(cid:90) d k α µν ( k ) (cid:101) g µν ( k ) (cid:19) where α (cid:48) is related to the string length, ( a, b ) denote the two-dimensional world-sheet coordinates, and X µ is the standard space-time coordinate. This similarity is of course not accidental because exponentiatingthe vertex operator in string theory does create a coherent state of gravitons! Our construction here isfrom eleven-dimensional point of view, and therefore more generic because of the absence of world-sheet inM-theory. – 43 – µν h / ( y, x ) as one would have expected. The other terms may be derived by computing thepath integral carefully. As in (2.71), the numerator of (2.77) takes the following form: (cid:90) [ D g µν ] e iS D † ( α ) δg µν D ( α ) = (cid:32)(cid:89) k (cid:90) d ( Re (cid:101) g µν ( k )) d ( Im (cid:101) g µν ( k )) (cid:33) exp (cid:34) iV (cid:88) k k | (cid:101) g µν ( k ) | + iS sol + ... (cid:35) × exp (cid:34) V (cid:88) k (cid:48) (cid:16) Re α ( ψ ) µν ( k (cid:48) ) Re (cid:101) g µν ( k (cid:48) ) + Im α ( ψ ) µν ( k (cid:48) ) Im (cid:101) g µν ( k (cid:48) ) (cid:17) + ... (cid:35) × V (cid:88) k (cid:48)(cid:48) ψ k (cid:48)(cid:48) ( x , y, z ) e − ik (cid:48)(cid:48) t (cid:0) Re (cid:101) g µν ( k (cid:48)(cid:48) ) + i Im (cid:101) g µν ( k (cid:48)(cid:48) ) (cid:1) exp (cid:32) − V (cid:88) k (cid:48) | α ( ψ ) µν ( k (cid:48) ) | (cid:33) , (2.78) where S sol is the action for the solitonic background (2.4) (plus the contributions from theG-fluxes that we will specify in the next sub-section), and δg µν is the part of g µν withoutthe solitonic piece. Comparing (2.71) with (2.78), we see that there are few differences.The action integral in (2.78) does not terminate to the quadratic term because of thepresence of interactions. However the simplified form of the kinetic term hides an infinite set of interactions with the soliton itself. These interactions arrange themselves to forma time-independent Schr¨odinger equation with a highly non-trivial potential. In fact thisis exactly the Schr¨odinger equation whose wave-functions are ψ k ( x , y, z ) discussed earlierand which appears in the Fourier decomposition (2.6)! Thus this wave-function appears inthe third line above replacing the e − i k (cid:48)(cid:48) · x in (2.71).The interactions, denoted by the dotted terms in the second and the third lines of(2.78) are important. They are classified by g | c | s M dp where ( c, d ) ∈ (cid:0) Z , Z (cid:1) , the 1 / g s <<
1, the path integral (2.78) can be exactly evaluatedand the result is: (cid:104) g µν ( x, y, z ) (cid:105) α = η µν h / ( y, x ) + Re (cid:32)(cid:90) d k ω ( ψ ) k α ( ψ ) µν ( k , t ) ψ k ( x , y, z ) (cid:33) + O (cid:18) g cs M dp (cid:19) = η µν (cid:16) Λ | t | √ h (cid:17) / + O (cid:32) g | c | s M dp (cid:33) + O (cid:34) exp (cid:32) − g / s (cid:33)(cid:35) , (2.79)which is exactly the M-theory uplift (2.4) of the de Sitter metric (at least the space-timepart of it), and matches well with what we had earlier in (2.16) using the Glauber-Sudarshanwave-function (2.13), and (2.17). One could also work out the expectation values of allthe internal components of the metric, namely g mn , g αβ and g ab , from the path integral as(2.78) and show that they match, up to corrections of O (cid:16) g | c | s / M dp (cid:17) , with (2.4). We willnot pursue this here: it’s a straight-forward exercise and may be easily performed. Insteadwe will discuss other immediate questions related to the path integral (2.78). Before goinginto this, note two things: one, the integral in (2.79) is over d k and not d k . This isbecause the kinetic term in (2.78) creates a pole at k = ω ( ψ ) k , and once we take the residueat that pole the dk integral goes away. Two, any extra terms, other than what appearsin (2.79), exactly cancel out from the denominator of (2.77), as one might have expected.– 44 –nother important point has to do with the O (cid:16) g cs M dp (cid:17) corrections in (2.79). For c > g s , but become non-perturbative when c <
0. Such non-perturbative series in inverse powers of g s may be resummed as a resurgent trans-seriesto take the expected non-perturbative form exp (cid:18) − g / s (cid:19) . This means when g s << small . The string coupling(which is the IIA string coupling) is given by g s √ h ( y ) = Λ | t | , and therefore as long as: − √ Λ < t ≤ , (2.80)we are at weak coupling. This rather awkward choice of the temporal domain results fromour type IIB metric (2.2) that allows a flat-slicing of de Sitter where −∞ < t ≤
0. We caneasily see that once we venture beyond the domain (2.80), both the perturbative and thenon-perturbative terms go out of control and we lose the simple classical feature (2.79). Theshifted interacting vacuum D ( α ( t )) | Ω( t ) (cid:105) no longer evolves as a coherent state, not evenapproximately, and the system truly becomes quantum (or at least we lose quantitativecontrol over it). It might be interesting to see if there are other expansion parameters, forexample a S-dual description in the IIB side, that could shed light for t < − √ Λ . However amore pertinent question is: what happens in the domain (2.80)? Do the O (cid:16) g cs M dp (cid:17) correctionsprovide small but finite contributions, or do they just cancel out ?Answering these questions will require us to introduce the other players in the field,namely the G-fluxes and quantum corrections. We will introduce them shortly, althoughnot before we analyze few other details pertaining to the path integrals discussed above.The first has to do with the fluctuations over the coherent state background. How do westudy them using path integrals?The analysis will involve a careful manipulation of the operator (2.46), because wehave interpreted the fluctuations over the de Sitter coherent state as the Agarwal-Tara [36]state (2.47). The operator (2.46), on the other hand, cannot have free Lorentz indices, soin the field formalism one way to express this would be to use even powers of the gravitonfield i.e. even powers of g µν g µν with no sum over the repeated indices. Thus what we arelooking for is an expectation value of the form: (cid:104) g µν (cid:105) Ψ ( c c ≡ (cid:104) Ω | D † ( α ( t )) (cid:16) c ∗ + c ∗ G † ( ψ ) ( a, a † ; t ) (cid:17) g µν (cid:16) c + c G ( ψ ) ( a, a † ; t ) (cid:17) D ( α ( t )) | Ω (cid:105)(cid:104) Ω | D † ( α ( t )) ( c ∗ + c ∗ G † ( ψ ) ( a, a † ; t )) ( c + c G ( ψ ) ( a, a † ; t )) D ( α ( t )) | Ω (cid:105) (2.81)= (cid:82) [ D g µν ] e iS D † ( α ) (cid:12)(cid:12)(cid:12) c + c (cid:82) d k (cid:80) m C ( ψ ) m ( k )( (cid:101) g µν ( k ) (cid:101) g µν ( k )) m (cid:12)(cid:12)(cid:12) g µν ( x, y, z ) D ( α ) (cid:82) [ D g µν ] e iS D † ( α ) (cid:12)(cid:12)(cid:12) c + c (cid:82) d k (cid:80) m C ( ψ ) m ( k )( (cid:101) g µν ( k ) (cid:101) g µν ( k )) m (cid:12)(cid:12)(cid:12) D ( α ) where the first line is expressed using operators and the second line is expressed usingfields. With some abuse of notation we have used D ( α ( t )) to denote the operator and D ( α ) The result (2.79) deviates from an exact de Sitter background of (2.4), albeit in a very small way,implying option (3). While this also justifies the point raised in footnote 26, there does exist a particularchoice of D ( α ) for which these extra correction terms do not appear thus implying option (2) instead. Tosee this we need to develop the story a bit more, and will be elaborated in section 3.3. – 45 –o denote the field. The latter is defined as in (2.76). We have also chosen a specific formof the G ( ψ ) function, as an integral over all momenta modes; and c i are the coefficients thatappear in (2.49) which, in turn, are bounded by (2.54). Note that the factor of ω ( ψ ) k + k does not appear explicitly. This is in fact redundant and is therefore absorbed in thedefinition of C ( ψ ) m ( k ) (see footnote 31). The coefficient C ( ψ ) m ( k ) is related to the coefficient C ( ψ ) m k ( t ) in (2.46).The path integral expression in (2.81) is rather complicated because it involves aninfinite sum of powers of the metric factor, so naturally a simplifying scheme is warrantedfor. We will start by making c = 1 and C ( ψ ) m ( k ) = δ m C ( ψ ) ( k ), as a toy example to seewhat kind of answer we expect from the path integral. With this in mind, the numeratortakes the following form: Num (cid:2) (cid:104) g µν (cid:105) Ψ ( c c (cid:3) = (cid:32)(cid:89) k (cid:90) d ( Re (cid:101) g µν ( k )) d ( Im (cid:101) g µν ( k )) (cid:33) exp (cid:34) iV (cid:88) k k | (cid:101) g µν ( k ) | + iS sol + ... (cid:35) × exp (cid:34) V (cid:88) k (cid:48) (cid:16) Re α ( ψ ) µν ( k (cid:48) ) Re (cid:101) g µν ( k (cid:48) ) + Im α ( ψ ) µν ( k (cid:48) ) Im (cid:101) g µν ( k (cid:48) ) (cid:17) + ... (cid:35) × V (cid:88) k (cid:48)(cid:48) ψ k (cid:48)(cid:48) ( x , y, z ) e − ik (cid:48)(cid:48) t (cid:0) Re (cid:101) g µν ( k (cid:48)(cid:48) ) + i Im (cid:101) g µν ( k (cid:48)(cid:48) ) (cid:1) exp (cid:32) − V (cid:88) k (cid:48) | α ( ψ ) µν ( k (cid:48) ) | (cid:33) , × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) c V (cid:88) k (cid:48)(cid:48)(cid:48) C ( ψ ) ( k (cid:48)(cid:48)(cid:48) ) (cid:2)(cid:0) Re (cid:101) g µν ( k (cid:48)(cid:48)(cid:48) ) + i Im (cid:101) g µν ( k (cid:48)(cid:48)(cid:48) ) (cid:1) (cid:0) Re (cid:101) g µν ( k (cid:48)(cid:48)(cid:48) ) + i Im (cid:101) g µν ( k (cid:48)(cid:48)(cid:48) ) (cid:1)(cid:3) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (2.82) that still involves a complicated set of nested integrals (which we express as sum over( k.k (cid:48) , k (cid:48)(cid:48) , k (cid:48)(cid:48)(cid:48) ) assuming discrete momenta). The way we have expressed (2.82), all thealternate lines involve real quantities. Even without doing anything we see that somestructure is evolving from the integrals. For example, when c = 0, we reproduce (2.79).This is no surprise. Now taking only the quartic term from the last line of (2.82), we seethat the numerator takes the following form: Num (cid:2) (cid:104) g µν (cid:105) Ψ ( c c (cid:3) = (cid:89) k (cid:32) πe − α ( k ) / k k (cid:33) (cid:34) η µν h / ( y, x ) + Re (cid:32)(cid:90) d k ω ( ψ ) k α ( ψ ) µν ( k , t ) ψ k ( x , y, z ) (cid:33)(cid:35) (2.83)+ (cid:89) k (cid:32) πe − α ( k ) / k k (cid:33) c (cid:20)(cid:90) d k (cid:48) k (cid:48) C ( ψ ) ( k (cid:48) ) α ( ψ ) µν ( k (cid:48) ) ψ k (cid:48) ( x , y, z ) e ik (cid:48) t (cid:18)
15 + α ( k (cid:48) )4 k (cid:48) + 5 α ( k (cid:48) ) k (cid:48) (cid:19) + ...... (cid:21) where the infinite factor in front comes from the gaussian integrals , and in the secondline we see that the term is suppressed by c because of the condition (2.54) imposed earlier.The dotted terms in the bracket involve other powers of the metric integrals from the lastline in (2.82), including the O (cid:16) g cs M dp (cid:17) corrections coming from the interaction Hamiltonian.We have also defined: α ( k ) ≡ α ( ψ ) µν ( k ) α ( ψ ) µν ( k ) , (2.84) This can in-fact be brought in a more manageable form Q ≡ exp (cid:20)(cid:82) + ∞−∞ d k log (cid:18) πe − α k ) / k k (cid:19)(cid:21) , butsince this will cancel out eventually, we won’t have to evaluate it. – 46 –nd similarly α ( k ). The pole structure of the second line is more complicated, as one wouldhave expected, and in addition there appears the factor C ( ψ ) ( k (cid:48) ) which is the remnant of thesimilar coefficient in (2.46). The denominator, to the same order in the graviton expansion,takes the following form: Den (cid:2) (cid:104) g µν (cid:105) Ψ ( c c (cid:3) = (cid:89) k (cid:32) πe − α ( k ) / k k (cid:33) (cid:20) c (cid:90) d k (cid:48) k (cid:48) (cid:18) α ( k (cid:48) )4 k (cid:48) + 3 α ( k (cid:48) ) k (cid:48) (cid:19) C ( ψ ) ( k (cid:48) ) + ....... (cid:21) , (2.85) where as before, the dotted terms appear from two sources, one, from the higher orderterms in the graviton expansion, and two, from the O (cid:16) g cs M dp (cid:17) corrections. Expectedly, thepole structures of the k integrals are different here, but the c suppression is consistentto what we had before. There is no appearance of the space-time wave-function ψ k ( x, y, z )and the Lorentz invariance is perfectly maintained. Dividing (2.83) by (2.85), we get: (cid:104) g µν (cid:105) Ψ ( c c = η µν h / ( y, x ) + Re (cid:32)(cid:90) d k ω ( ψ ) k α ( ψ ) µν ( k , t ) ψ k ( x , y, z ) (cid:33) + c (cid:90) d k f µν ( k , k ) ψ k ( x , y, z ) e ik t + ... = η µν (cid:16) Λ | t | √ h (cid:17) / + c (cid:90) d k f ( k , k ) α ( ψ ) µν ( k , k ) ψ k ( x , y, z ) e ik t + O (cid:32) g | c | s M dp (cid:33) + O (cid:34) exp (cid:32) − g / s (cid:33)(cid:35) , (2.86) where we see that the infinite factors in front of (2.83) and (2.85) have cancelled out.The second line of (2.86) tells us the Agarwal-Tara state [36], or the GACS (2.49) indeedreproduces the fluctuations over our de Sitter space, realized as a Glauber-Sudarshan state.The Fourier coefficient f ( k , k ) appearing above is now defined as: f ( k , k ) = C ( ψ ) ( k )8 k (cid:18)
15 + α ( k )4 k + 5 α ( k ) k + ... (cid:19) + O ( c ) + O (cid:18) g cs M dp (cid:19) , (2.87)which should be compared to (2.25), (2.26) and (2.27). The pole structures in (2.87) aredifferent from (2.27), which is expected because (2.27) is for IIB and (2.87) is for M-theory.Nevertheless, the awkward factors of T that renders (2.27) somewhat non-convergent, donot appear in (2.87). This means, the way we have expressed our fluctuations, (2.86) doprovide an answer to the Trans-Planckian issue, namely:The time-dependent frequencies that we encounter from fluctuations over a de Sitter vac-uum are actually artifacts of Fourier transforms over a de Sitter state , viewed as a Glauber-Sudarshan state.There are however two questions that may arise in interpreting (2.86) as fluctuations overde Sitter space. The first is the form of f ( k , k ) given in (2.87) and in (2.27). There appearsto be α ( ψ ) µν ( k ) dependence in (2.87) whereas none appears in (2.27). The reason is simple:the form (2.87) is an integrand, and once we insert the functional form for α ( ψ ) µν ( k ) from(2.9), and sum over higher values of m , we should be able to reproduce (2.27). Howeversince (2.87) is always a function of k , awkward factors like T may not appear in the final– 47 –xpression. This suggests that the path-integral is much more powerful way to analyze theexpectation values here.The second question is the form of C ( ψ ) m ( k ) in (2.46). Will it create new poles in (2.87)?The answer depends on the way we construct the Agarwal-Tara state (2.49) and (2.46).The coefficient C ( ψ ) m ( k ) is the Fourier transform of C ( ψ ) m k ( t ) from (2.46), so depending on ourchoice of (2.46) there could be higher order zeroes or poles. However this could be fixedfrom the very beginning, so that the integral (2.87) may determine the eventual behaviorof f ( k , k ).Now that we have understood how to interpret fluctuations over the de Sitter state, itis time to discuss the second issue, namely the existence of the Wilsonian effective action.In other words we would like to ask the following question: how do the background (2.4)get affected once we integrate out the momentum modes from a cut-off Λ UV to say M p ?To proceed, and as it goes for the Wilsonian integration procedure, we will have to goto the Euclidean formalism. We will also define the field configuration, as given in (2.18),to the following: g µν ( x, y, z ) = η µν h / ( y, x ) + (cid:90) +M p − M p d k (cid:101) g µν ( k , k ) ψ k ( x , y, z ) e − ik t (2.88)+ (cid:90) − M p − Λ UV d k (cid:101) g µν ( k , k ) ψ k ( x , y, z ) e − ik t + (cid:90) +Λ UV +M p d k (cid:101) g µν ( k , k ) ψ k ( x , y, z ) e − ik t , where all dynamics bounded by | k | < M p are used to define the effective field theory.The modes lying between M p < | k | ≤ Λ UV are integrated out, but now we see that thisprocedure can indeed be explicitly performed because the modes themselves have no trans-Planckian issues. We will start by asking how this effects the action S , the action thatappears in say (2.78). In fact all we need is (cid:104) α | α (cid:105) which appears in the denominator of(2.78) and is defined as: (cid:104) α | α (cid:105) ≡ (cid:82) [ D g µν ] e iS D † ( α ) D ( α ) (cid:82) [ D g µν ] e iS , (2.89)which is not identity, as we saw earlier, because D ( α ) is not unitary. The reason why(2.89) suffices is because, the expectation value of the space-time metric in (2.78) may bere-written as the following integral: (cid:104) g µν ( x, y, z ) (cid:105) α = 2 lim Λ UV →∞ (cid:90) +Λ UV − Λ UV d k ψ k ( x , y, z ) e ik t ∂∂α ∗ ( ψ ) µν ( k ) log (cid:20) exp (cid:18)(cid:90) +Λ UV − Λ UV d k (cid:48) | α ( ψ ) ( k (cid:48) ) | (cid:19) (cid:104) α | α (cid:105) (cid:21) , (2.90) where the | α | part is defined in the same way as in (2.84) but now with α ( ψ ) µν and itscomplex conjugate. The integrals are bounded by the cut-off Λ UV which, here, we assumewould approach infinity although it could in principle be any scale. The LHS is a physicalquantity so it’s scale dependence could be subtle. However on the RHS, once we changethe scale, other quantities would change accordingly, for example the action S will become S eff and all the fields would get re-scaled, to keep the zeroth order (in g s ) result, i.e. (2.79)– 48 –or g s →
0, unchanged. We will see later, once we do explicit computations, that this isindeed the case . There are also G-fluxes whose dynamics we haven’t discussed yet. The solitonic solution(2.1) require non-trivial G-fluxes to allow for the compact internal eight-manifold M in(2.3). These G-fluxes are time-independent quantities that that have non-trivial functionalbehavior over the eight-manifold (2.3), including components along the 2 + 1 dimensionalspace-time itself. If (M , N) denotes components in the internal eight-manifold, and ( µ, ν )the components along the space-time directions, then we require flux components of theform G MNPQ ( y ) and G µνρ M ( y ) to allow for the solitonic background like (2.1) to exist (see[41, 34, 43, 44] for the special case when h ( y, x ) = h ( y )). Supersymmetry is preservedin the special case when the internal G-fluxes are self-dual over the eight-manifold (2.3)[41]. Our present aim is to search for fluxes that can go along with a background like (2.4).Clearly now we are looking for coherent states, or Glauber-Sudarshan wave-functions, thatmay capture the specific fluxes or flux components as most probable values (or alterna-tively, as expectation values). Two questions follow here: one, how do we construct thecorresponding Glauber-Sudarshan wave-functions? And two, what are the consistencyconditions that allow such states to exist in the first place?From the start we know that all flux components have to be time- dependent so thatcoherent state representations may at least be constructed. This is however easier said thandone: one needs to overcome numerous subtleties to even start attempting a constructionof this kind. For example, fluxes in a compact manifold have to be quantized, otherwisewe will face problems with the Dirac quantization procedure. If now the fluxes becometime-dependent, how do we even justify that there is a quantization procedure? Even moreseriously, what would such quantization procedure mean: quantization at every instant oftime? Or more like quantization over some interval of time?Even if we find some meaning to the quantization procedure, the fluxes have to satisfyGauss’ law , otherwise consistent construction cannot even be attempted. Here we wantthe fluxes to be time-dependent. How do we satisfy Gauss’ law? Do we want Gauss’ lawto be satisfied at every instant of time or, as again before, more like over an interval?The subtleties don’t end here. Time-dependent fluxes have to break supersymmetriesso that we could consistently support a non-supersymmetric Glauber-Sudarshan state thatprovide the metric configuration (2.4) as expectation value (see (2.79)). Breaking super-symmetry in either the metric or the flux sectors will switch on an uncontrolled plethoraof time-dependent quantum corrections. How do we control them?Surprisingly the answer to all these questions come from an unexpected corner: fromthe large plethora of time-dependent quantum corrections! These have been answered ingreat details in [21] so we will be brief here. Both the flux quantization and the Gauss’ law This however does not immediately imply that there is no scale dependence. In fact this is also tiedup to the question of the renormalization of the four-dimensional Newton’s constant. We will discuss thistowards the end of section 3.3. Sometime also called the anomaly cancellation condition. – 49 –re related to the Bianchi identities and the G-flux EOMs. As shown in sections 4.2.1 and4.2.2 of [21], these set of equations get corrected, order by order in g cs / M dp , from the infinitetowers of quantum corrections (both local and non-local). The question then is: how dothese corrections help us to balance the flux-quantization and Gauss’ law when the fluxes,as well as the underlying manifold, are varying with time?The answer, as shown in [21] is as simple as it is instructive. The plethora of quantumcorrections are known to scale in very specific ways with respect to time , or here, withrespect to g s /H where H ( y ) = h ( y ) = h ( y ). On the other hand, we have claimed that theG-fluxes themselves are time-dependent, i.e. they too scale with respect to g s /H in veryspecific ways (that we will elaborate below). This means, in either Bianchi identities orEOMs, augmented by the quantum corrections, all we need is to identify equivalent powersof g s /H from flux components and from the quantum corrections! For flux quantizations,we need some integrated form of the identities (see details in [21]), but the moral of thestory should be clear: order-by-order in g s /H equations could be balanced and one couldgive meaning to both flux-quantizations and anomaly-cancellations in a time-dependentbackground.However, how do we know that such balancing of the g s /H terms is consistent with theEOMs? The answer to this, as shown in [21], is again rather simple. The flux-quantizationprocedures, or the integrated forms of the Bianchi identities, are in-fact the EOMs of the dual-forms , i.e. the EOMs of the seven-form G MNPQRST that are Hodge-dual of thefour-form G MNPQ . Thus the very act of balancing the equations order-by-order in g s /H ,we are in-fact solving the dual seven-form EOMs! Such construction is then self-consistentto solving the four-form EOMs, implying an overall self-consistency of the system in thepresence of time-dependent degree of freedom.Our aim then is to re-interpret a part of the story as appearing from the Glauber-Sudarshan wave-functions. For that we will require similar Fourier modes as in (2.6) whichwe got from the metric configurations (2.1) and (2.4). However new subtleties appearregarding what to choose: the three-form fields C MNP or the four-form G-fluxes G MNPQ to study the dynamics? The reason is, if we choose the three-form to be g s dependent(or time-dependent), then this will inadvertently lead to a four-form G-flux components ofthe form G . Such components will break de Sitter isometries in the type IIB side socannot be allowed. How can we find a way to not allow such components to arise here?The answer, as shown in section 4.2.3 of [21], again lies in the quantum corrections. Inthe absence of M5-branes, the EOM for a seven-form flux component becomes the Bianchiidentity of the corresponding four-form G-flux component. Solving this will genericallygive G = d C + quantum corrections, implying that G is not just d C but comes withan additional baggage of quantum terms. This freedom can be used to set: G = 0 , (2.91) Recall we have identified g s with time t via the following relation g s H = √ Λ | t | . Note also that H = H ( y )denotes the warp-factor whereas H denotes the Hubble parameter. The Hubble parameter H will onlyfeature prominently in section 4 whereas the warp-factor H = H ( y ) appears in section 2 and 3. The bold faced flux components either denote the g s dependent G-fluxes from [21, 22], or fields/operatorshere. To avoid confusion, only one notation will be used throughout. – 50 –hus saving us from breaking the four-dimensional de Sitter isometries in the type IIB side.Therefore it appears either C MNP or G MNPQ may be used to study the background fromM-theory.It turns out, unfortunately that this is not quite true. There is yet another level ofsubtlety that we have kept under the rug and it has to do with G-flux components like G MN ab where ( M, N ) ∈ M × M and ( a, b ) ∈ T G as in (2.3). Such components clearlyviolate the type IIB de Sitter isometries, unless: G MN ab ( y m , y α , y a , g s ) ≡ F MN ( y m , g s ) ⊗ Ω ab ( y α , y a , g s ) , (2.92)where Ω ab is a localized two-form on M × T G and F MN appears as type IIB gauge field onseven-branes. How and why such seven-branes appear have been explained in [21, 22], sowe will avoid going into it here. Instead we will point out two aspects of (2.92). The firstone is the obvious one: the decomposition does not produce a three-form field, rather aone-form gauge field. Such a gauge field can become non-abelian, but that’s another storythat we will avoid getting into . The second one is not very obvious: the g s dependences of F MN and Ω ab . It turns out there is a natural way to impart both g s and M p dependence toΩ ab that come from the metric (2.4) along the toroidal directions. As a first approximationthen we can keep F MN to be g s independent.Our discussion above might have convinced that a uniform description with G-fluxesappear once we take field strengths and not fields. The field strengths are also gaugeinvariant and in many cases they capture quantum corrections that neither the three-form C MNP nor the one-form A M captures. It would then appear that the coherent states maybe constructed for the G-flux components directly. In fact this would fit well with theEOMs themselves, as EOMs involve only field strengths and not fields themselves [21].Typically then we expect: (cid:101) G MNPQ ( k ) = (cid:90) d x (cid:88) p,n ≥ ( − δ n G ( p,n )MNPQ ( y ) (cid:16) g s H (cid:17) p/ exp (cid:32) − nH / g / s (cid:33) h / h γ k ( x, y, z ) , (2.93) where G (0 , ( y ) is the self-dual G-flux that we switch on to support the solitonic back-ground (2.1). The sum is over ( p, n ) and typically p ∈ Z with p ≥ as shown in [21]. Near g s →
0, the dominant contributions from the non-perturbative piece come from n = 0, andtherefore the higher order G-flux contributions, alluded to earlier, appear from G ( p, ( y )for p ≥ . The other important ingredient of (2.93) is the Schr¨odinger wave-function γ k ( x, y, z ) that should be compared to the wave-functions appearing in (2.7), and notably ψ k ( x, y, z ). A similar wave-function also appears in the Fourier decomposition of the fol-lowing G-flux components: (cid:101) G µνρ M = (cid:90) d x (cid:15) µνρ ∂ M (cid:20) h ( y ) (cid:18) | t | − (cid:19)(cid:21) h / h β k ( x, y, z ) , (2.94) Here it will suffice to say that wrapped M2-branes on vanishing two-cycles of the internal eight-manifold(2.3) play an important role in making the system non-abelian. For more details see [45]. – 51 –ith the corresponding Schr¨odinger equation for β k ( x, y, z ). As mentioned earlier, thetemporal behavior of these wave-functions with spatial form γ k ( x , y, z ) and β k ( x , y, z ),would be captured for the corresponding coherent state wave-functions with frequencies ω ( γ ) k and ω ( β ) k respectively.There are a few subtleties that we should point out regarding the constructions (2.93)and (2.94). The first has to do with the corresponding fields C MNP and C µνρ ≡ C ij .Modulo the issues mentioned earlier, we could also expand them in powers of g s /H , andtake their Fourier transforms in terms of two other set of wave-functions. However sincethe three-form fields are not gauge invariants, there would be issues when we want to study fluctuations over the solitonic background. In the path-integral analysis this will introduceFaddeev-Popov ghosts, thus complicating the ensuing analysis. To avoid this we will stickwith the wave-functions γ k ( x , y, z ) and β k ( x , y, z ), keeping in mind that they represent thefield-strengths and therefore differ somewhat with the other set of wave-functions (2.7).Note that such an argument also extends to the G-flux components of the form (2.92),although there appears a possibility that k < for such cases (see details in [22]).The second subtlety has to do with the Glauber-Sudarshan wave-function. What isthe form of the wave-function now? The answer remains the same as before: it is relatedto the shifted interacting vacuum (2.55). In other words, we expect the configuration spacewave-function (2.56), much along the lines of (2.65), to still capture the essential dynamicsof the system, except now the interaction Hamiltonian H int include interactions with the G-fluxes too. The displacement operator D ( α ) in (2.61) will naturally become more involvedbecause the annihilation and the creation operators a eff and a † eff respectively will involvethe flux sector too. The interacting vacuum is still annihilated by a eff , but the rest ofthe construction simply gets more involved retaining, however, the essential features thathelped us to cement the coherent state construction. Expectation values over the coherentstate, for example (cid:104) G MNPQ (cid:105) α and (cid:104) G ij M (cid:105) α , would provide the required time- dependent fluxes to support a background like (2.4).Finally, how is the supersymmetry broken? The answer, in the language of the expec-tation values, is simple. We demand: (cid:68) G MNPQ (cid:16) G MNPQ − ( ∗ G ) MNPQ (cid:17)(cid:69) σ (cid:54) = 0 , (2.95)where ∗ is the Hodge dual of the four-form over the solitonic internal metric (2.1) and σ is the generalized coherent states that we will elaborate in section 3.3. The un-warpedinternal metric is a non-K¨ahler one (including a non-K¨ahler metric on the base M × M of (2.3)), and the non-zero value on the RHS of (2.95) comes from the quantum termsalluded to earlier. A generic derivation of (2.95) is a bit technical, so we will only discussit later in section 3.3. The readers could also refer to [3, 46] and [21] for details. Or more generically (cid:104) G MNPQ (cid:105) ( α,β ) and (cid:104) G ij M (cid:105) ( α,β ) as we shall explain in section 3.3. An alternative, and probably more useful, way to express supersymmetry breaking is to demand |(cid:104) G MN ab (cid:105) σ − (cid:104) ( ∗ G ) MN ab (cid:105) σ | > , N) ∈ M × M and ( a, b ) ∈ T G (as takenfrom (2.3)). The reason for choosing this specific component of G-flux is because of it’s appearance in theSchwinger-Dyson’s equations (see section 3.3 for details). – 52 – . Quantum effects and the Schwinger-Dyson equations We have by now more or less specified all the essential ingredients that go in the construc-tion of the Glauber-Sudarshan state, namely the metric and the G-flux components. Allthe time-dependent quantities, for example the metric (2.4), appear as expectation valuesover the Glauber-Sudarshan states. What we haven’t specified yet are the actual form ofthe quantum terms that go in the interaction Hamiltonian H int in (2.55). In the followingtwo sections, 3.1 and 3.2, we will revisit and hence elaborate further how these quantumterms shaped our construction of the time-dependent background (2.4), and the corre-sponding G-flux components from (2.93) and (2.94). This means the metric and the G-fluxcomponents entering the discussion in sections 3.1 and 3.2 are those of the metric (2.4)and the corresponding g s dependent flux components, and not of the solitonic background(2.1) (the bold-faced symbols signify those, so should not be confused with operators inthe earlier sections). In section 3.3 we will see how the results of the sections 3.1 and 3.2appear from the Glauber-Sudarshan states. This is where the Schwinger-Dyson equationswould become useful. Our starting point would be the perturbative series of quantum effects that include bothlocal and non-local terms. These quantum terms will in turn determine the O (cid:16) g cs M dp (cid:17) cor-rections to the Glauber-Sudarshan wave-function from (2.65). In the following we keep c > d could take any signs. All of these are of course embedded inside the eleven-dimensional action that we write in the following way: S = M p (cid:90) d x √− g (cid:16) R + G ∧ ∗ G + C ∧ G ∧ G + M p C ∧ Y (cid:17) (3.1)+ (cid:88) { l i } ,n i (cid:90) d x √− g (cid:32) Q T ( { l i } , n , n , n , n )M σ ( { l i } ,n i ) − p (cid:33) + M p ∞ (cid:88) r =1 (cid:90) d x √− g c ( r ) W ( r ) − n T (cid:90) d σ (cid:26)(cid:112) − γ (2) (cid:16) γ µν (2) ∂ µ X M ∂ ν X N g MN − (cid:17) + 13 (cid:15) µνρ ∂ µ X M ∂ ν X N ∂ ρ X P C MNP (cid:27) , where in the first line we denote the standard kinetic terms and the interactions in M-theory,in the second line we denote other possible interactions, and in the third line we introduce n number of integer and fractional M2-branes . The M p scalings of each of the termsare denoted carefully, including the ones that involve complicated interactions (see (3.10)below). These interactions will involve polynomial powers of the curvature tensors, G-fluxcomponents and possible derivative actions. To quantify them we will have to get downto more finer space-time notations, advocated early in the sections. For example if ( m, n )denote the coordinates of M , ( α, β ) the coordinates of M and ( a, b ) the coordinates of Here we are simply adding the action of the individual M2-brane assuming well separation betweenthem. When they are on top of each other such a simple addition is not possible and one should look forsomething more along the lines of the Bagger-Lambert form [47]. Such construction will make the systemeven more complicated than it already is, so we will avoid it here. – 53 – G in (2.3), the quantum term Q T ( { l i } , n , n , n ) may be expressed as: Q ( { l i } ,n i )T = g m i m (cid:48) i .... g j k j (cid:48) k { ∂ n m }{ ∂ n α }{ ∂ n a }{ ∂ n } ( R mnpq ) l ( R abab ) l ( R pqab ) l ( R αabβ ) l × ( R αβmn ) l ( R αβαβ ) l ( R ijij ) l ( R ijmn ) l ( R iajb ) l ( R iαjβ ) l ( R mnp ) l × ( R m n ) l ( R i j ) l ( R a b ) l ( R α β ) l ( R αβm ) l ( R abm ) l ( R ijm ) l × ( R mnpα ) l ( R mαab ) l ( R mααβ ) l ( R mαij ) l ( R mnα ) l ( R m α ) l ( R αβα ) l × ( R abα ) l ( R ijα ) l ( G mnpq ) l ( G mnpα ) l ( G mnpa ) l ( G mnαβ ) l ( G mnαa ) l × ( G mαβa ) l ( G ijm ) l ( G ijα ) l ( G mnab ) l ( G abαβ ) l ( G mαab ) l , (3.2)which is written completely in terms of all the possible non-zero components of curvaturetensors and G-flux components that would appear from the background (2.4). The sum isover all l i and n i , so this would be an exhaustive collections of all possible interactions inthe system. However one might question the absence of other possible interactions. Aren’tthey important? The answer is: not here, because as we shall see in section 3.3, these areexactly the interactions that would enter the Schwinger-Dyson equations which would helpus to express the EOMs as expectation values over the Glauber-Sudarshan states. In theWilsonian analysis that we did earlier, the other components could be thought of as beingintegrated out. Interestingly, in the limit ( n , n , n ) → ( ∞ , ∞ , ∞ ) the interactions startto become non-local . In the next set of interactions in (3.2), we consider a more advancedform of non-local interactions using W ( r ) ( y ) which are nested integrals of the form: W ( r )( { l i } ,n i ) ( y ) = M p (cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) F ( r ) ( y − y (cid:48) ) W ( r − { l i } ,n i ) ( y (cid:48) ) (3.3)= M p (cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) F ( r ) ( y − y (cid:48) ) (cid:90) d y (cid:48)(cid:48) (cid:112) g ( y (cid:48)(cid:48) ) F ( r − ( y (cid:48) − y (cid:48)(cid:48) ) W ( r − { l i } ,n i ) ( y (cid:48)(cid:48) ) . with r and F ( r ) ( y − y (cid:48) ), the latter depending implicitly on ( g s , M p ), denoting the level ofnon-localities and the non-locality functions respectively with F (0) ( y − y (cid:48) ) ≡ c ( r ) are numerical constants, and the lowestorder non-local interaction W (1) ( y ) may be expressed in terms of (3.2) as: W (1)( { l i } ,n i ) ( y ) ≡ (cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) (cid:32) F (1) ( y − y (cid:48) ) Q ( { l i } ,n i )T ( y (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) . (3.4)The list of interactions mentioned above may not still be the full set of perturbative interac-tions that theory could allow for a generic choice of g s , but for g s < non-perturbative effects although clearly higher order perturbative interactions from M2 and M5 instan-tons are captured by (3.2), and so are the effects from the wrapped branes in (3.3). The The limit n → ∞ is non-locality in time and is discussed as case 4 in section 3.2.6 of [21]. This issurprisingly harmless. We could also absorb n by shifting n , n and n . Typically D-branes and instantons in type II theories contribute terms of order g s and g s respectively.The perturbative contributions then come from the higher-order world-volume terms that scale as g θs with θ >
0. In a charge neutral configuration, the factor of 2 in the exponent off-sets the inverse g s factors,and therefore they can contribute to (3.1). – 54 –opological corrections are assimilated in: Y ≡ a tr R + a (cid:0) tr R (cid:1) + a (cid:0) tr R (cid:1) (cid:0) tr G (cid:1) + a tr G + a (cid:0) tr G (cid:1) + ..., (3.5)with ( a , a ) taking numerical values while ( a , a , a , .. ) proportional to powers of M p so that (3.5) remains dimensionless. The dotted terms are additional possible traces.All the parameters appearing above are two-forms that should not be confused with thecorresponding tensors .Another interesting thing to note is the ubiquity of four-form G-fluxes in the action(3.1), although there are a few places where the three-form C MNP appears. For most ofthese cases we could alternatively use the four-form fluxes to rewrite in the following way: (cid:90) M G ∧ G ∧ G = (cid:90) ∂ M C ∧ G ∧ G , (cid:90) M C ∧ Y = − (cid:90) M G ∧ Y , (3.6) where the eleven-dimensional space-time M ≡ R (1 , × M × M × T G is consideredas a boundary of a twelve -dimensional space-time M . This clearly suggests a F-theoryuplift of our construction, as, such a coupling do appear there (see [48]), connecting in turndirectly to the type IIB dual description.For the second case in (3.6), we are assuming that Y is a locally exact form d Y . Inthe limit a = a = a = 0 in (3.5), and with appropriate choice of ( a , a ) (see [21]),the locally exact form is easy to demonstrate. Once we switch on G-flux components, weexpect similar feature to show up. However the three-form C MNP do appear as the chargeterm for the M2-branes, so we will still need to deal with this. Additionally the EOMsfor the G-fluxes are variations of the action (3.1) with respect to the three-form, so itwould be a worth-while exercise to express the three-form in terms of Fourier componentsusing the Schr¨odinger wave-functions ( γ (cid:48) k , β (cid:48) k ), much like ( γ k , β k ) used in (2.93) and (2.94)respectively. The connections between ( γ (cid:48) k , β (cid:48) k ) and ( γ k , β k ) are not so straightforward andmay be formally presented as: γ (cid:48) k ≡ γ (cid:48) k ( γ k , β k , Ψ k ) , β (cid:48) k ≡ β (cid:48) k ( γ k , β k , Ψ k ) , (3.7)thus mixing with both ( γ k , β k ) and even involving the other set of Schr¨odinger wave-functions Ψ k as in (2.7) from the gravitational sector. Fortunately however, as we shall seea bit later, we will not have to deal with this here.The energy-momentum tensors from the interacting part of the action (3.1) are nowimportant. It is also important to keep track of all the space-time directions carefully so Here we define the two-form in the standard way from the corresponding curvature tensors and G-fluxcomponents using the vielbeins e a o P and the holonomy matrices M a o b o , as: R ≡ R a o b o MN M a o b o dy M ∧ dy N , G ≡ G a o b o MN M a o b o dy M ∧ dy N R a o b o MN ≡ R MNPQ e a o P e b o Q , G a o b o MN ≡ G MNPQ e a o P e b o Q . By construction R is dimensionless, but G has a dimension of length so to make it dimensionless we needto insert M p . This in fact will determine the M p scalings of ( a , a ) in (3.5). Note that the Wilson surfacewill become exp (cid:0) i M p (cid:82) C (cid:1) , and so would be the charge of the M2-branes. The latter is captured by T in(3.1). – 55 –e will follow the finer subdivision, namely ( m, n ) , ( α, β ) and ( a, b ) for the internal space(2.3), and ( µ, ν ) for the remaining 2 + 1 dimensional space-time. Let us now define thefollowing variables: N ≡ (cid:88) i =1 l i , N ≡ n + n + n − n + l + l N ≡ l + l + l ) , N ≡ l + l + l , N ≡ l + l + l ) , (3.8)where l i are the powers of the curvature tensors and G-flux components appearing in(3.2); and n i , with i = (0 , , ,
3) are the derivatives along the temporal, M , M andthe toroidal directions T G respectively. We can typically take n = 0. This way the fieldsremain independent of the toroidal directions and therefore do not jeopardize the dualityto the IIB side. With this we can define the energy-momentum tensor from the quantumterms along the (M , N) ∈ (cid:16) ( m, n ) , ( α, β ) (cid:17) directions in the following way: C ( q, p )MN ≡ − √− g ∂ S int ∂ g MN = (cid:88) { l i } ,n i σ ( { l i } ,n i ) p (cid:32) − g MN Q ( { l i } ,n i )T + ∂ Q ( { l i } ,n i )T ∂ g MN (cid:33) (3.9) × δ (cid:16) N + N + ( p + 2) N + (2 p + 1) N + ( p − N − q − (cid:17) , where for a given value of q ≥ p ≥ /
2, the delta function above gives an equationin terms of N i , or alternatively, in terms of ( l i , n i ). The integer solutions of these are thensummed over to provide the full contributions from the quantum corrections. The M p scalings of each of these quantum terms, given here and in (3.1) by σ ( { l i } , n i ), also getfixed because: σ ( { l i } , n i ) ≡ N + N + N + 12 ( N + N ) , (3.10)showing that all the quantum terms entering in (3.9) have in general different M p scalings.So at the face value there is a M p hierarchy in the quantum terms. When p = 0, welose the g s hierarchy completely because there are an infinite number of terms for anyvalues of q ≥ C ( q, mn . What about M p hierarchy? From (3.10) we do however seem toretain the M p hierarchy, although a careful consideration with the localized fluxes (2.92)as discussed in section 2.2 of [22], show that this is not true. Since the p = 0 case is thetime- independent case from (2.93), it suggests that there are an infinite number of quantumcorrections with neither g s nor M p hierarchies contributing to the system. This is a clearsign of a breakdown of an effective field theory (EFT) description in the time-independentcase (see also [21, 22] for more details).In analyzing the energy-momentum tensor for the quantum term (3.9) we have ignoredthe non-local terms. They can be easily accommodated in because the non-local counter-terms (3.3) and (3.4) are defined using nested integrals from (3.2) and using non-localityfunctions. These functions tend to become very small at low energies (see section 3.2.6of [21]), so we can ignore their contributions. Nevertheless, even if we do take their con-tributions into account, as done in section 3 of [22], the problems associated with p = 0case do not get alleviated, implying that an EFT description cannot be restored in the– 56 –rientation g s scaling IIB dual M × M g s Taub-NUT instanton along R , M × T G g s D3 instanton along M M × ( S ) α × ( S ) a =3 1 g s NS5 instanton along M × ( S ) α × ( S ) a =3 M × ( S ) α × ( S ) b =11 1 g s D5 instanton along M × ( S ) α × ( S ) a =3 Table 1:
The g s scalings of the wrapped M5-instantons along various six-cycles inside the non-K¨ahler eight-manifold (2.3) in M-theory; with ( α, β ) parametrizing the coordinates of M and( a, b ) parametrizing the coordinates of T G . The last two configurations, wrapped on local one-cycles, break the four-dimensional de Sitter isometries in the IIB side so cannot contribute to theenergy-momentum tensors. time-independent case. Once time-dependences are switched on, i.e. when p ≥ /
2, anEFT description with a finite set of local and non-local quantum terms and well-defined( g s , M p ) hierarchies, magically appear.The energy-momentum tensor along the toroidal direction, i.e. along T G , has a similarstructure to (3.9) but with some minor, and crucial, changes. It may be expressed as: C ( q, p ) ab ≡ − √− g ∂ S int ∂ g ab = (cid:88) { l i } ,n σ ( { l i } ,n i ) p (cid:32) − g ab Q ( { l i } ,n i )T + ∂ Q ( { l i } ,n i )T ∂ g ab (cid:33) (3.11) × δ (cid:16) N + N + ( p + 2) N + (2 p + 1) N + ( p − N − q + 4 (cid:17) , with p still bounded below by p ≥ /
2, but now, more importantly, q ≥
6. This keepsthe last two terms negative definite, which is what we want for the system to make sense.This awkward factor of 6 may be easily explained from the metric configuration (2.4) thatscales as g / s along the toroidal direction. Finally, the energy-momentum tensor for thespace-time directions differ from (3.9) and (3.11) in the following way: C ( q, p ) µν ≡ − √− g ∂ S int ∂ g µν = (cid:88) { l i } ,n σ ( { l i } ,n i ) p (cid:32) − g ab Q ( { l i } ,n i )T + ∂ Q ( { l i } ,n i )T ∂ g µν (cid:33) (3.12) × δ (cid:16) N + N + ( p + 2) N + (2 p + 1) N + ( p − N − q − (cid:17) , with the same lower bound on p as before, and we expect q ≥
0. There are however a fewsubtleties that need to be elaborated on before we can fix the value of q . This is what weturn to next. Before we derive the subtleties associated with the energy-momentum tensor along the 2+1dimensional space-time, C ( q, p ) µν , we need to re-visit the local and the non-local quantumterms to search for non-perturbative effects that can contribute. Recall that the non-perturbative effects were Borel summed to exp (cid:18) − g / s (cid:19) , so in the limit g s →
0, they– 57 –imply decouple. Is this always true? In the following we want to argue that this may notalways be true and there could be terms that do contribute.
Our starting point is the perturbative series of quantum terms in (3.2). We can ask whetherwe can generalize this further. For example, how about powers of the individual terms in(3.2)? The answer is that it not necessary to do this because: Q ( { l i } ,n i )T ( y ) ⊗ Q ( { l j } ,m j )T ( y ) ≡ Q ( { l i + l j } ,n i + m j )T ( y ) , (3.13)so in principle (3.2) does capture the most generic perturbative quantum effects, and anyarbitrary modifications to it is already embedded in the series itself. This is of course anexpected property of the underlying renormalization group itself, so its appearance hereshould not be of any surprise.However such a generalization unfortunately do not extend to the non-local counter-terms. The non-local counter-terms are expressed in terms of nested integrals (3.3), and wedo not expect products of two nested integrals could produce another equivalent integral.For example: W ( r i )( i ) ( y ) ⊗ W ( r j )( j ) ( y ) = M p (cid:90) d y (cid:48) d y (cid:48)(cid:48) (cid:112) g ( y (cid:48) ) g ( y (cid:48)(cid:48) ) F ( r i ) ( y − y (cid:48) ) F ( r j ) ( y − y (cid:48)(cid:48) ) W ( r i − i ) ( y (cid:48) ) W ( r j − j ) ( y (cid:48)(cid:48) ) (cid:54) = W ( r i + r j )( i + j ) ( y ) ≡ M p (cid:90) d z (cid:112) g ( z ) F ( r i + r j ) ( y − z ) W ( r i + r j − i + j ) ( z ) , (3.14) with W ( r i )( i ) ( y ) = W ( r i )( { l i } ,n i ) ( y ); which remains true unless the non-locality function, givenby F ( r ) ( y − z ), becomes a localized function of the form δ ( y − z )M p √ g ( z ) . In the latter case thisreduces to (3.13). Taking (3.14) and (3.13) into account suggests that there could beadditional quantum terms of the form: U ( r )( { l i } ,n i ) ( y ) = ∞ (cid:88) n =1 d n M np (cid:18)(cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) F ( r ) ( y − y (cid:48) ) W ( r − { l i } ,n i ) ( y (cid:48) ) (cid:19) n = ∞ (cid:88) n =1 ( − n d n M np (cid:18)(cid:90) d z (cid:112) g ( y − z ) F ( r ) ( z ) W ( r − { l i } ,n i ) ( y − z ) (cid:19) n , (3.15)for any choice of the level of non-locality r . In the second equality we simply redefined thecoordinated to shift the non-localities to the metric and the quantum terms. At the lowestlevel of non-locality, i.e. for r = 1, we can simplify (3.15) in the following way: U (1)( { l i } ,n i ) ( y ) ≡ ∞ (cid:88) n =1 d n (cid:34)(cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) (cid:32) F (1) ( y − y (cid:48) ) Q ( { l i } ,n i )T ( y (cid:48) )M σ ( { l i } ,n i ) − p (cid:33)(cid:35) n (3.16)where we have restricted the coordinate dependences only on the base M × M of (2.3),and therefore both the determinant of the metric and the M p scaling change accordingly.The absence of the warped torus volume V T , that appeared prominently in [21], is im-portant to get the scaling right. We will also take Ω ab = (cid:15) ab in (2.92). Thus one should– 58 –nterpret (3.16) as a separate class of non-local interactions. Later however we will con-sider the case where the torus volume does show up. The other term appearing above is Q ( { l i } ,n i )T ( y (cid:48) ) which is given in (3.2). The g s scaling of the n -th term in the series expansionmay be written as g θ p s where θ p is given by: θ p = n (cid:16) N + N + (2 + p ) N + (2 p + 1) N + ( p − N − (cid:17) , (3.17)with N i defined as in (3.8) and F (1) ( y − y (cid:48) ) do not explicitly depend on g s . Since p ≥ / N (cid:54) = 0 by switching on quantum terms associated with the G-flux components G MN ab in(3.2), the quantum term (3.16) blows up in the limit g s →
0. This is because of the g s factor from the determinant of the metric in (3.16). Studying the EOMs order by order inpowers of g s , as shown in [21, 22], we switch on smaller values of N , N and N and forthese values (3.16) tends to blow-up. However such a series, for appropriate choices of d n ,could be summed as a trans series to take the following form : U (1) ( y ) ≡ (cid:88) { l i } ,n i ,k c k exp (cid:34) − k (cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) (cid:32) F (1) ( y − y (cid:48) ) Q ( { l i } ,n i )T ( y (cid:48) )M σ ( { l i } ,n i ) − p (cid:33)(cid:35) (3.18) ≈ (cid:88) { l i } ,n i ,k c k exp (cid:34) − k (cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) (cid:32) F (1) ( − y (cid:48) ) Q ( { l i } ,n i )T ( y (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) + k O (cid:18) y ∂ F (1) ( − y (cid:48) ) ∂y (cid:48) (cid:19)(cid:35) , where the validity of the second line rests on the smallness of the derivative of the function One concern is whether this can always be done. If d n = ( − n n ! then there is no doubt that (3.18) isalways true with c k = 0 for k >
1. Question is what happens in the generic case. Generically however wecan expect: d n ≡ n ! ∞ (cid:88) l =1 c l ( − l ) n = ( − n c n ! + ( − n c n ! + ( − n c n ! + .... where d n is defined in (3.15) and c l are positive or negative integers. We can now go to the limit where theterms inside the bracket in (3.16) are smaller than 1. Plugging this in say (3.15), every term of d n , whensummed over n , will produce an exponentially decaying contribution like (3.18) when the integrals thereinbecome much bigger than 1. Solutions would exist if the following matrix: M = − − − − .... .... − − − − ....
124 23 278 323 ....... .... ... has an inverse. Unfortunately M is infinite dimensional so in practice it will be impossible to ascertainthe full inverse of such a matrix. However since higher values of c l and n also correspond to smaller andsmaller contributions, it would make sense to terminate M to large but finite dimensions. For such cases,one may verify that the inverses continue to exist thus giving more practicality to the series of d n above.Additionally, assuming wide separation between two consecutive c i and c j , the dominant term will alwaysbe the first few terms of (3.18). Relatively, therefore it makes sense to keep only the first few terms of(3.18) as others will die-off faster than this when g s → θ <
2, for every choice of ( { l i } , n i ). When θ >
2, one may perturbatively expand the exponential to arbitrary orders and study the corresponding g s scalings. – 59 – (1) ( − y (cid:48) ). Interestingly, in this limit, the first term looks suspiciously close to the actionof a gas of neutral BBS M5-instantons [37] wrapped on the base M × M of our eight-manifold (2.3) once we absorb the function F (1) ( − y (cid:48) ) in the quantum series (3.2) (see Table 1 ). The tell-tale sign of g s from the determinant of the base metric, which signalsthe presence of the wrapped instantons, in fact deters it to contribute to the energy-momentum tensor unless the g s scaling from Q ( { l i } ,n i )T ( y (cid:48) ) for any choices of ( { l i } , n i ) isalways bigger than 2. For the two energy-momentum tensors that we derived in (3.9)and (3.11), the lowest order quantum corrections contributing to the EOMs scale as g / s [21, 22]. For these cases, even though the non-local counter-terms like (3.3) do contribute,there appears to be no contribution from (3.18). Can higher order in g s contribute? Letus infer it from the following quantitative analysis, without using any approximations: T (np;1)MN ( z ) = − p (cid:112) − g ( z ) δδ g MN ( z ) (cid:18)(cid:90) d x (cid:112) − g ( x ) U (1) ( x ) (cid:19) (3.19)= (cid:88) { l i } ,n i ,k c k g MN ( z ) exp (cid:34) − k (cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) (cid:32) F (1) ( z − y (cid:48) ) Q ( { l i } ,n i )T ( y (cid:48) )M σ ( { l i } ,n i ) − p (cid:33)(cid:35) + (cid:88) { l i } ,n i ,k kc k M σ ( { l i } ,n i ) − p (cid:90) d x F (1) ( x − z ) δ (cid:16)(cid:112) g ( z ) Q ( { l i } ,n i )T ( z ) (cid:17) δ g MN ( z ) (cid:115) g ( x ) g ( z ) × exp (cid:34) − k (cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) (cid:32) F (1) ( x − y (cid:48) ) Q ( { l i } ,n i )T ( y (cid:48) )M σ ( { l i } ,n i ) − p (cid:33)(cid:35) , where we have used the fact that δ g MN ( w ) δ g PQ ( z ) = dp δ MP δ NQ δ d ( w − z ), with no extra √ g d factorwith d = 11 ,
6, the latter because we have assumed the internal coordinates y (cid:48) = ( y m , y α )to only span the coordinates of the base M × M of eight-manifold (2.3). The first termof (3.19) is interesting: its the metric component suppressed by an exponentially decayingfactor (for small g s scaling of (3.2)). We will not worry about this term as we shall showlater that such a term is cancelled by a counter-term. The second term, on the otherhand, contains the quantum pieces (3.2), again suppressed by the exponential factor. It isinstructive to note the g s scalings of every components of the energy-momentum tensor: T (np;1) ab ( z ) = (cid:88) k c k (cid:20) g / s + kg s (cid:16) g θ +4 / s (cid:17)(cid:21) exp (cid:18) − kg s · g θs (cid:19) T (np;1) µν ( z ) = (cid:88) k c k (cid:20) g − / s + kg s (cid:16) g θ − / s (cid:17)(cid:21) exp (cid:18) − kg s · g θs (cid:19) T (np;1) mn ( z ) = T (np;1) αβ ( z ) = (cid:88) k c k (cid:20) g − / s + kg s (cid:16) g θ − / s (cid:17)(cid:21) exp (cid:18) − kg s · g θs (cid:19) , (3.20)where θ ≡ n ( θ p + 2 n ) and θ p as in (3.17). Notice that the only way (3.20) can contribute iswhen θ ≥
2. From our earlier studies in [21, 22], θ ≥ / a, b ) , ( m, n )and ( α, β ), and therefore naively there seems to be no contributions from their correspond-ing energy-momentum tensors (3.20). The only energy-momentum tensor that appears tocontribute seems to be T (np;1) µν ( z ). Close, but not exactly the same: the integral structure here is a bit different from an actual BBSinstanton gas contribution. – 60 –he above conclusion is not quite correct once we look at the case corresponding to θ = 8 / a, b ) , ( m, n ) and ( α, β ). The four components now scale as( g s , g − s , g s , g s ) with the adjoining exponential pieces going as exp (cid:16) − kg / s (cid:17) for all of them.Recall from [21] and [22], these g s scalings are exactly how the classical EOMs scale, andtherefore it appears that there are non-perturbative corrections that scale as θ = 8 /
3, withthe corresponding energy-momentum tensor going as: T (np;1)MN ( z ) = (cid:88) { l i } ,n i ,k c k exp (cid:34) − k (cid:90) d y (cid:48) (cid:112) g ( y (cid:48) ) (cid:32) F (1) ( − y (cid:48) ) Q ( { l i } ,n i )T ( y (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) + k O (cid:18) ∂ F (1) ( − y (cid:48) ) ∂y (cid:48) (cid:19)(cid:35) (3.21) × g MN ( z ) + 2 k M σ ( { l i } ,n i ) − p (cid:90) d x F (1) ( − z ) δ (cid:16)(cid:112) g ( z ) Q ( { l i } ,n i )T ( z ) (cid:17) δ g MN ( z ) (cid:115) g ( x ) g ( z ) + ... , once we impose the approximation of slowly varying F (1) ( − y ). Our exact expression,(3.19), tells us that this approximation is not necessary but is useful nevertheless because itappears to isolate the instanton effects from all the non-local factors. The problem howeverwith this approximation is the appearance of the eleven-dimensional volume factor for thesecond term in (3.21), unless we use a box normalization condition. Clearly this issue doesnot arise in (3.19) where the eleven-dimensional volume factor is regulated by the function F (1) ( x − z ), for example as: (cid:90) d x (cid:112) − g ( x ) F (1) ( x − z ) = g − / s F e ( z ) , (3.22)where F e ( z ) is a well-defined finite function even if we did not include the contributionsfrom the exponential factor. The latter would further improve the value of the function,so its exclusion from (3.22) does not change anything (see (3.57)). The interesting take-home point from such an integral is that the effects of the non-localities from F (1) ( z − y (cid:48) ), F (1) ( x − z ) and F (1) ( x − y (cid:48) ) in (3.19) are effectively removed by the nested integrals sothat the final result for the energy-momentum tensor is a consistent quantity. Expandingperturbatively the exponential factors in (3.19), we get: T (np;1)M a N a ( z ) = g l a s (cid:34) − g / s (cid:88) k kc k + O ( g / s ) (cid:35) + g / l a s g s (cid:88) k (cid:104) kc k (1 − kg / s ) + O ( g / s ) (cid:105) , (3.23) where the subscript a specifies the components, with l a taking the corresponding values,in (3.20). We have also expanded the exponential piece exp (cid:16) − kg / s (cid:17) perturbatively inpowers of g / s . The integer k can be arbitrarily large and let us assume that it approachesinfinity as k → (cid:15) − with (cid:15) →
0. In that case as long as g s goes to zero as g s → (cid:15) b , with b bounded by: 32 < b < (cid:12)(cid:12)(cid:12)(cid:12) − (cid:18) log | c max | log (cid:15) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (3.24)– 61 –nd c max being the largest value of c k , most of the series appearing above should beconvergent, except the one without g s suppression. Such a coefficient can in principle beabsorbed in the definition of F (1) ( x − z ) which we have left unspecified so far .It is important to note that we have imposed one condition on θ to (3.9) and (3.11),and a different condition on θ to the corresponding components in (3.19). The reasonis simple: although the quantum series appearing in both these set of energy-momentumtensors are the same, i.e. the series (3.2), the latter has an extra 1 /g s suppression goingwith it, creating the necessary difference in the outcome. If we go beyond r = 1 and sum thetrans-series in (3.15), much like how we did before , the contributions to the EOMs fromany of the corresponding energy-momentum tensors are very small. The generic actionmay be expressed as: S = (cid:88) { l i } ,n i ,k (cid:90) d x √− g ∞ (cid:88) r =1 c k exp (cid:104) − k M p (cid:90) d y (cid:112) g ( y ) F ( r ) ( x − y ) W ( r − ( y ; { l i } , n i ) (cid:105) , (3.29) where we again assume dependence on the coordinates of M × M of the internal eight-manifold (2.3) although, as we shall discuss in section 3.2.3, the warped toroidal volume We have used (3.24) to allow for a convergent series by restricting ourselves to O ( g / s ). One can doslightly better than this by noting that the series in k may be bounded in the following way: (cid:88) k kc k < k ( k + 1) | c max | < k | c max | (cid:88) k k c k < k ( k + 1)(2 k + 1) | c max | < k | c max | , (cid:88) k k n c k < k n +1 | c max | , (3.25)which determines the generic bound that one could place on the perturbative expansion of the exponentialterm. To see what value of n should suffice, we note that the perturbative expansion of the exponentialfactor from (3.19), to order n typically involve a term of the form: S ( z ) = (cid:88) k c k k n +1 n ! (cid:32) F (1) ( w − w ) g / s M σ ( i ) − p (cid:33) n ≡ (cid:88) k c k k n +1 n ! Γ n ( w , w ) , (3.26)which is heavily suppressed by various factors like g s , M p and F (1) ( w − w ) including n ! so long as σ ( i ) > σ ( i ) ≡ σ ( { l i } , n i ) is defined in (3.10). Such suppression factors tell us that we need not go beyondsome order of expansion for the exponential part in (3.19). Thus if we go up to order p , then it is easy tosee that the bounds in (3.24) become:32 < b < p (cid:12)(cid:12)(cid:12)(cid:12) p + 2 − (cid:18) log | c max | log (cid:15) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (3.27)which reproduces (3.24) as a special case when p = 1. To see what happens for arbitrary values of ( k, n ),let us assume that k goes to a large value k max ≡ (cid:15) − . We can now use the lower bound on b from (3.27)to first sum over k in (3.26) and then sum over n . In this case, it is clear that the series in (3.26) may bebounded from above by: S ( z ) < | c max | k exp (cid:16) − k max Γ ( w , w ) (cid:17) , (3.28)which is how we can control the series right to the point where g s goes to zero as g s = (cid:15) b , with b as in(3.27). Thus for (cid:15) b < g s <
1, both | c max | and k max can be arbitrarily large, yet S ( z ) in (3.26) can still be finite . Implying that the same conditions used earlier apply here too. – 62 –ould be inserted in the integrand. The action that appeared in (3.19) is the restrictivepiece with r = 1, with no additional dependence on the toroidal volume. The other functionappearing above is defined in (3.8). So far we discussed various ways of generating non-perturbative terms that contribute as1 /g s to the energy-momentum tensors. From Tables 1 and it seems other wrappedinstantons either do not contribute − by breaking the de Sitter isometries in the IIBside − or they just contribute perturbatively, as positive powers of g s . What about typeIIB seven-branes wrapped on M ? These seven-branes would map to Taub-NUT spacesoriented along M × T G i.e. along ( α, β ) and ( a, b ) directions. We could even include moregeneric seven-branes that are not necessarily D7-branes. Their dual, in M-theory, wouldbe warped Taub-NUT spaces whose properties are not too hard to ascertain (see [49, 45]).Unfortunately two obstacles forbid a naive realization of this scenario: one, a Taub-NUTspace cannot have a simple product geometry as M × T G ; and two, we cannot allow non-trivial axio-dilaton charge in the type IIB side, as this will change the type IIA couplingcompletely, ruining our basic g s scaling behavior.The only way out is to allow for a charge neutral configuration of the seven-branes inthe IIB side. In fact this is exactly the F-theory scenario with 24 seven-branes wrapping M and stretched along the 3 + 1 dimensional space-time. The orthogonal space to theseven-branes is a P with 24 points where the F-theory fibre torus degenerates [50]. Thisspace could be identified with M , allowing us to realize such a configuration from M-theory. In fact one could even go a step further: write an equivalent seven-dimensionalaction using the normalizable forms on the base manifold, that involves the higher orderquantum terms in the following way : S = (cid:88) { l i } ,n i T (cid:90) d σ √− g g ab ∂ a ∂ b (cid:32) ˆ Q ( { l i } ,n i )T ( y, y α , y a )M σ ( { l i } ,n i ) − p (cid:33) , (3.30)where the quantum terms are similar to the ones that we encountered in (3.2), the differencenow being their dependence on the coordinates of the fibre torus. Concerns about thecharges of the seven-branes are no longer there because of the charge neutrality in theIIB side. Additionally, in the weak-coupling limit, (3.30) reproduces the higher orderaction on the IIA six-branes. Expectedly a T-duality along x direction then provides us aconfiguration of space-filling, but charge neutral, seven-branes in the IIB side that do notbreak any of the de Sitter isometries. Clearly the gauge fields on the seven-brane wouldbe generated from the M-theory G-flux component via a decomposition similar to (2.92),where Ω ab ( y a , y α ) is now a function of the M × T G coordinates.The inclusion of g ab and the derivatives along the toroidal directions tell us that weare now taking n = 2 in (3.8) (which we had put to zero earlier). The fact that this Except for a class of instantons, that we will discuss a bit later. As it happens in the usual case, the coordinates of the eight-manifold (2.3) are represented in (3.30)by: y ≡ y ( σ ) , y α ≡ y α ( σ ) and y a ≡ y a ( σ ). We can find a gauge where σ M is identified with the coordinatesof R , × M . – 63 –omes with a negative sign in (3.8) can be turned to our advantage here. To see this weshall first assume that the time dependences of the gauge field F MN and the two-formΩ ab can be collected together as (cid:0) g s H (cid:1) p/ so that it remains consistent with the generictemporal-dependences advocated in (2.93) or in [21]. This way:Ω ab ( x ) = ∞ (cid:88) n =1 B n exp (cid:0) − M np x n (cid:1) (cid:15) ab , (3.31)where B n are M p and g s independent constants. Note also the absence of any g s dependentfactors as they have already been accounted for. One could in principle deviate from thisto allow for a slightly different option where the g s dependence appear in the exponent, asconsidered in [22], but then this forms a different class of solution that we will discuss abit later. Putting everything together, the energy-momentum tensor from (3.30) becomes: T (np;2 a )MN ( z ) = − (cid:88) i T (cid:112) − g ( z ) (cid:90) d y (cid:48) M σ ( i ) − p (cid:112) g ( y (cid:48) ) g bb ∂ b · δδ g MN ( z ) + δ (cid:16)(cid:112) − g ( y (cid:48) ) g bb (cid:17) δ g MN ( z ) ∂ b ˆ Q ( i )T ( y, y b ) , (3.32) where the sum is over all ( { l i } , n i ) and y b ≡ x ( y (cid:48) ), thus g bb = g , . Note that we haverestricted the dependence on the coordinates of M and x , but a more generic dependencewith the coordinates of M should not be too hard. The simplified dependence helps us toexpress the world-volume metric by g ( y (cid:48) ); and ∂ b depends on the embedding y b = y b ( y (cid:48) )via: ∂ b = (cid:32) ∂y (cid:48) P ∂y b ∂y (cid:48) Q ∂y b (cid:33) ∂ ∂y (cid:48) P ∂y (cid:48) Q . (3.33)If we take B n = 0 for n > p so that this terms scales with respect to M p in the same way as (3.19). Regarding the g s scalings, the various components of the energy-momentum tensor scale in the followingway: T (np;2 a ) mn ( z ) = T (np;2 a ) αβ ( z ) = 1 g s (cid:16) g θ − / s (cid:17) T (np;2 a ) µν ( z ) = 1 g s (cid:16) g θ − / s (cid:17) , T (np;2 a ) ab ;2 ( z ) = 1 g s (cid:16) g θ +4 / s (cid:17) , (3.34)similar to the second terms in (3.19), with θ being the same as the one appeared there. In-terestingly, when θ = 8 /
3, the four set of energy-momentum tensors scale as ( g s , g s , g − s , g s ),with no relative suppressions between them as we had for (3.19). However their contri-butions rely crucially on the embedding y b = y b ( y (cid:48) ), and therefore also on the derivativeconstraint ∂ b ˆ Q ( i )T ( y, y b ) (cid:54) = 0. Choosing a standard embedding of the seven-branes wouldmake these contributions vanish.Although the seven-branes themselves don’t seem to contribute non-perturbativelyto the energy-momentum tensor, the world-volume fields on the seven-branes in princi-ple could. One specific set of contributions could come from the fermionic terms on the– 64 –rientation g s scaling IIB dual S ∈ M g s D3 instanton along (cid:0) S ∈ M (cid:1) × ( S ) a =3 (cid:0) S ∈ M (cid:1) × ( S ) α g s D3 instanton along (cid:0) S ∈ M (cid:1) × ( S ) α × ( S ) a =3 ( S ) m × M g s D3 instanton along ( S ) m × M × ( S ) a =3 M × ( S ) a =3 g s D1 instanton along M (cid:0) S ∈ M (cid:1) × ( S ) a =3 g s D1 instanton along S ∈ M M × ( S ) b =11 g s F1 instanton along M (cid:0) S ∈ M (cid:1) × ( S ) b =11 g s F1 instanton along S ∈ M ( S ) α × T G g / s Kaluza-Klein instanton along ( S ) α ( S ) m × T G g / s Kaluza-Klein instanton along ( S ) m Table 2:
The g s scalings of the wrapped M2-instantons along various three-cycles inside the non-K¨ahler eight-manifold (2.3) in M-theory; with ( m, n ) parametrizing the coordinates of M , ( α, β )the coordinates of M and ( a, b ) the coordinates of T G . Again most cases break the four-dimensionalde Sitter isometries in the IIB side so they cannot contribute to the energy-momentum tensors.Some of the above configurations rely on the existence of local one-cycles inside the non-K¨ahlersub-manifolds. Absence of these will remove their contributions further. seven-branes. In M-theory we should then look for possible fermionic completions of thequantum terms like (3.2). Fermionic terms imply introducing the Gamma matrices, andsince two Gamma matrices anti-commute to the corresponding metric components, theGamma matrices themselves should become time- dependent (because the metric compo-nents are). What does that mean?It means that the eleven-dimensional Gamma matrices should be expressed in termsof the eleven-dimensional vielbeins as Γ M ≡ Γ ¯ a e ¯ a M and e ¯ a M e ¯ a N = g MN , where Γ ¯ a are thestandard constant Gamma matrices that now anti-commute to the flat metric η ¯ a ¯ b . Thefermionic completion of the four-form G-flux in M-theory, may be expressed using anti-symmetric products of Gamma functions and fermions, somewhat along the lines of [51],in the following way:ˆ G MN ab ( y m , y α , y a , g s ) ≡ e ¯ Υ P Γ MNPQ ab Υ Q ( y, g s ) + e ¯ Υ [M Γ ab Υ N] ( y, g s ) , (3.35)where Υ M is the eleven-dimensional gravitino and e i are just constants. Despite its com-plicated appearance compared to what we encountered in [51], due solely to the presenceof eleven-dimensional gravitino, this is not new (see for example [52]). Once we decompose Υ M ( y, g s ) = Ψ ( y m , g s ) ⊗ Θ M ( y α , y a , g s ) + Ψ M ( y m , g s ) ⊗ Θ (cid:48) ( y α , y a , g s ), then, compared to(2.92), we have the following forms: ˆΩ ab ∝ ¯Θ (cid:48) Γ ab Θ (cid:48) , ˆΩ MN ab ∝ ¯Θ [M Γ ab Θ N] , ˆΩ M ab ∝ ¯Θ [M Γ ab ] Θ (cid:48) (3.36) – 65 – Ω (cid:48) MN ab ∝ ¯Θ P Γ MNPQ ab Θ Q , ˆΩ (cid:48) MNQ ab ∝ ¯Θ P Γ MNPQ ab Θ (cid:48) , ˆΩ (cid:48) MNPQ ab ∝ ¯Θ (cid:48) Γ MNPQ ab Θ (cid:48) , alongwith ¯ˆΩ M ab and ¯ˆΩ MNQ ab ; which would make them functions of ( y m , y α , g s ) from thevielbeins. We expect the internal fermions and gravitinos, i.e. Θ and Θ N , to reproducea behavior like (3.31), from the corresponding Dirac and Rarita-Schwinger equations on M × T G . This is bit more complicated scenario so some simplification is warranted for. Inthe generic scenario we place no constraints on Θ and Θ N right now. With these in mind,let us consider the following variation of (2.92):ˆ G MN ab ( y m , y α , y a , g s ) = ¯ Ψ (cid:16) e ˆΩ MN ab + e ˆΩ (cid:48) MN ab (cid:17) Ψ + e ¯ Ψ [M ˆΩ ab Ψ N] + e ¯ Ψ [P ˆΩ (cid:48) MNPQ ab Ψ Q] + e ¯ Ψ ˆΩ [M ab Ψ N] + e ¯ Ψ ˆΩ (cid:48) MNQ ab Ψ Q + h . c , (3.37)where ( Ψ , Ψ M ) are the two kinds of fermions in M-theory, now localized along R , × M ,and e ij are constants. If we assume that the fermions do not have any g s dependences,then ¯ Ψ = Ψ † Γ = Ψ † Γ ¯ a e a , as well as ¯ Ψ M , would naturally scale as g / s . If we restrict(M , N) ∈ M , then Γ MN scales as g − / s , implying that the coefficients of e ij terms in (3.37)all scale as g / s . The G-flux component G MN ab in (2.92), on the other hand, scales as g k/ s .This mismatch has a natural explanation. In a time-dependent background, if weswitch on fermionic bilinears, they cannot be time-independent. The simplest bilinear func-tion will involve two fermions without any Gamma functions, implying that the fermionsthemselves should have some g s dependences. With this in mind, let us arrange for thefollowing g s dependences for the two kinds of fermions in M-theory: Ψ ( x , y m , g s ) = (cid:88) k (cid:48) Ψ (2 k (cid:48) +4) ( x , y m ) (cid:16) g s H (cid:17) k (cid:48) / , Ψ M ( x , y m , g s ) = (cid:88) k (cid:48) Ψ (2 k (cid:48) +4)M ( x , y m ) (cid:16) g s H (cid:17) k (cid:48) / , (3.38) which is arranged so that (3.37) or (3.35) transform exactly as the corresponding G-fluxcomponents G mnab , G αβab and G mαab when k (cid:48) = ( k − y α , y a ) and Θ M ( y α , y a ) do not have any g s dependences. We have also inserted a spatialdependence on x , to the already expected spatial dependence on y m , in anticipation of thefollowing decomposition: Ψ ( x , y m ) = ψ ( x ) ⊗ χ (4) ( y m ) , Ψ M ( x , y ) = ψ ( x ) ⊗ χ (4) m ( y m ) , (3.39)with m ∈ M and ψ i ( x ) being fermionic degrees of freedom in R . Clearly this decomposi-tion leads to variety of fermionic degrees of freedom but the problem arises when we try torestore the de Sitter isometries in the IIB side. One way out is to remove the x dependencealtogether and view the fermions to be completely on the sub-manifold M and considersimply the bilinear (3.37). On one hand, if we consider the bilinear ¯ ΨΨ ( y m , g s ), this wouldscale as (cid:0) g s H (cid:1) k (cid:48) + k (cid:48)(cid:48) +2) / , which seems to contribute for ( k (cid:48) , k (cid:48)(cid:48) ) > (0 , g s expansions for Ψ and Ψ M in (3.38) would make sense if k >
4. More importantlyhowever in the decomposition (3.37), the gravitino degrees of freedom Ψ M cannot appearbecause type IIB seven-branes do not have gravitino fields on their world-volumes. This– 66 –an be made possible if from the start we impose Θ (cid:48) ( y α , y a ) = 0. In that case there existsthe following fermionic completion of the four-form flux: G totMN ab ≡ G MN ab + M p ˆ G MN ab , (3.40)where the first term, from (2.92), provides the world-volume gauge fields in the IIB side;and the second term, from the first line of (3.37) with e i = e i = 0, provides the fermionicterms. Both these contributions provide U (1) degrees of freedom, but can be made non-abelian if we incorporate wrapped M2-branes. These non-abelian degrees of freedom willreside on multiple seven-branes at a point on M , and could therefore form a structuresimilar to the one discussed for the heterotic theories in [53]. We have also assumed that thefermion Ψ is dimensionless so that (3.40) remains dimensionless also. All these informationsmay be inserted in the following quantum action: S = (cid:88) { l i } ,n i ,q T (cid:90) d σ √− g (cid:16) ¯ ΨΨ (cid:17) q × (cid:32) (cid:101) Q ( { l i } ,n i )T ( y m , y α , g s )M σ ( { l i } ,n i ) − p (cid:33) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ r ≤ l r =0 , (3.41)where (cid:101) Q ( { l i } ,n i )T ( y m , y α , g s ) is the same one as in (3.2) except G MN ab therein is replacedby the fermionic bilinear (3.37). We could have instead replaced G MN ab by G totMN ab , butthen we have to keep track of the M p scalings a bit more carefully. Additionally wehave augmented the quantum piece with a bilinear without extra Gamma-functions. Thequantum terms in (3.41) provide us an expression with higher powers of curvature andthe fermionic bilinears, but there are no interactions with G-flux components (which inprinciple could be constructed by relaxing the l r = 0 constraints, or by inserting G totMN ab in (3.2)). If the g s scaling of the full quantum term in (3.41) is given by g θ k s , and the M p scaling by M σ ( { l i } ,n i ) p , then θ k and σ ( { l i } , n i ), take the following form: σ ( { l i } , n i ) = (cid:88) i =0 n i + N , θ k = 13 (cid:16) (cid:88) i =0 n i + N + ( k − N + 2 q ( k − (cid:17) , (3.42)with no dependence on N for σ ( { l i } , n i ), and we have used 2 k (cid:48) = k − p scalings ofthe actual G-flux components do depend on N . This can be rectified by giving a dimensionof M − / p to the fermion Ψ itself and zero dimensions to the internal fermions Θ and Θ M .Note that θ k , computed without incorporating contribution from the g in (3.41), wouldbe positive definite for k >
2. The other quantities, namely N i , n i , are defined in (3.8).The energy-momentum tensor from (3.41) becomes: T (np;2 b )MN ( z ) = − (cid:88) i,q T (cid:112) − g ( z ) (cid:90) d y (cid:48) (cid:0) ¯ ΨΨ (cid:1) q M σ ( i ) − p (cid:112) − g ( y (cid:48) ) δδ g MN ( z ) + δ (cid:16)(cid:112) − g ( y (cid:48) ) (cid:17) δ g MN ( z ) (cid:101) Q ( i,r )T ( y (cid:48) , y b ) , (3.43) which should be compared to what we had in (3.32). One thing to note is that while(3.32) depends crucially on the embedding of the seven-branes, (3.43) is independent of– 67 –he embedding. Once we insert the G-flux components, i.e. insert the anti-symmetrictensors along with their fermionic bilinears in (3.37), we can have gauge fields and fermionsinteracting with each other and contribution to the potential. Generalizations aside, the g s scalings of the various components of the energy momentum tensors may be written as: T (np;2 b ) mn ( z ) = T (np;2 b ) αβ ( z ) = 1 g / s (cid:16) g θ k − / s (cid:17) T (np;2 b ) µν ( z ) = 1 g / s (cid:16) g θ k − / s (cid:17) , T (np;2 b ) ab ( z ) = 1 g / s (cid:16) g θ k +4 / s (cid:17) , (3.44)with θ k as in (3.42). This contributes to the classical EOMs in [21] at θ k ≥ / (cid:0) ¯ Ψ Ω MN ab Ψ (cid:1) and other higher-order termsfor k ≥ / q = 0. However if we had taken q >
0, then k ≥ / n + n + N + q ) + 3 N = 8 , (3.45)which can allow terms like (cid:0) ¯ ΨΨ (cid:1) , amongst other possible contributions coming from thecurvature, gauge fields and their derivatives. They are finite in number because k > k = 0, and θ k = 4 / k = 0 is not the time-independent case for the fermions which may be seen fromour scaling (3.38) above.Before ending this section, let us clarify one worrisome feature related to the possibilityof vanishing contribution from a term like (cid:0) ¯ Ψ Ω MN ab Ψ (cid:1) and higher powers. This is muchlike what happens in the ten-dimensional case with gauge singlet Majorana-Weyl fermions[52]. However such concerns are alleviated once we go to the non-abelian case by includingwrapped M2-branes on vanishing cycles. In the dual IIB side these wrapped M2-branesbecome tensionless strings between coincident F-theory seven-branes. The non-abelianenhancement occurs geometrically via [54] or algebraically via the Tate’s algorithm [55].In the non-abelian case this would convert: (cid:0) ¯ Ψ Ω MN ab Ψ (cid:1) → (cid:0) tr adj ¯ Ψ Ω MN ab Ψ (cid:1) , (3.46)which is in general non-zero. Here we have taken the trace in the adjoint representation,although one could in principle construct it for arbitrary representation of the gauge group.In (3.40), the total G-flux G totMN ab then is comprised of the usual tensorial flux components G MN ab , and the traces over the fermionic bilinears instead of the abelian ones discussed in(3.37), much like what we know in the heterotic side from [53]. So far our concentrations have mostly been towards scenario where g s →
0. In this limitmany of the non-perturbative contributions vanish because of their dependence on either– 68 –xp (cid:18) − g / s (cid:19) or on exp (cid:16) − g s (cid:17) , the latter being from the wrapped instantons. What hap-pens for g s <
1, where some of the exponential factors do not go to zero as fast as it werewhen g s →
0? To study the implications of these, let us express (3.31) alternatively as (seealso [22]): Ω ab ( x , y α , g s ) = ∞ (cid:88) n,k =1 B nk exp (cid:16) − M np g n/ s x n (cid:17) exp (cid:104) − ( y α y α ) k g − k/ s M kp (cid:105) (cid:15) ab , (3.47) where y α y α = g αβ y α y β with un-warped metric g αβ . In [22] we studied the case where allB nk = 0 except B . Here we would like to concentrate on B (cid:54) = 0 in addition to B .Note the appearance of g s in the exponents themselves. This may seem to take us awayfrom the generic time-dependences of the G-flux components in (2.93), giving us: Ω ab ( x , y α , g s ) = B exp (cid:16) − M p g / s x (cid:17) (cid:34) B exp (cid:32) − M p y α g / s (cid:33)(cid:35) (cid:15) ab ≡ (cid:15) ab + B exp (cid:32) − M p y α g / s (cid:33) (cid:15) ab , (3.48) in the decomposition (2.92). The second equality appears from restricting the coordinatedependence to the base M × M . However such a choice of the two-form do not changethe g s scaling of (3.2) with θ = n ( θ p +2 n ), and θ p as in (3.17), unless the ( n , n ) derivativeactions in (3.2) act on (3.48). It is this action, in particular the one associated with n in(3.2), is what we are interested in here.To see how this develops, let us start with U (1) ( y ) as in (3.18), with two differences: one,the integral is over the six-manifold M × T G , and two, we choose (3.48), with (B , B ) (cid:54) =(0 , = 0. In the IIB side this will be instantons wrapping M (see Table1 ), and we will call these the delocalized
KKLT type instanton gas [5]. Thus the limit weare looking at here is: g s → (cid:15), M p → (cid:15) − / , T (np;1)MN ( z ) = T (np;1)MN ( z ; B = 1) , (3.49)with T (np;1)MN ( z ) is as given in (3.19) and (cid:15) <
1. Interestingly, in this limit, even if we hadentertained a non-zero B , the exponential factor would have gone to zero as exp (cid:0) − (cid:15) (cid:1) for (cid:15) →
0. When (cid:15) <
1, the coefficient of B doesn’t go to zero as fast. The energy-momentumtensor on the other hand, deviates from (3.19) in the following instructive way: T (np;3)MN ( z ) = (cid:88) { l i } ,n i ,k c k g MN ( z ) exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g n ) / s (cid:32) F (1) ( z − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) ˆ V (cid:35) + (cid:88) { l i } ,n i ,k kc k M n p V M σ ( { l i } ,n i ) − p (cid:90) d x ( − z α ) n F (1) ( x − z ) g n ) / s · δ (cid:16)(cid:112) g ( z ) Q ( { l i } ,n i )T ( z ) (cid:17) δ g MN ( z ) (cid:115) g ( x ) g ( z ) × exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g n ) / s (cid:32) F (1) ( x − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) ˆ V (cid:35) , (3.50) where the dotted terms are now the less dominant ones with powers of n that may bederived from (3.2). There are a few differences from (3.19), as shown in red above: one,– 69 –nstanton type V a V b T g s scaling T (np)MN θ min BBS a = b = 0 g θ + las g s (3.19) Delocalized BBS a = 0 , b = 1 g θ + las g / s (3.54) with n = 0 KKLT a = b = 0 g θ + l a s (3.56) with n = 0 Delocalized KKLT a = 1 , b = 0 g θ + las g / s (3.53) Table 3:
The g s scalings of the four kind of wrapped M5-instantons that contribute to the non-perturbative energy-momentum tensor T (np)MN . The two-form Ω ab that defines the G-flux componentsin (2.92), is chosen to be Ω ab = (cid:16) exp (cid:0) − M p y α (cid:1) (cid:17) (cid:15) ab , with B a constant independentof ( g s , M p ). The g s scalings remain unchanged for the two cases with B = 0 and B (cid:54) = 0,although their corresponding M p scalings change. The other parameter θ is defined in the text as θ ≡ n ( θ p + 2 n ) and θ p as in (3.17) with θ min being the minimum value of θ that contributes tothe non-perturbative energy-momentum tensors; l a = 4 / , − / , − / V is the unwarped volume of the sub-manifold M ; and V T is the unwarped volumeof the toroidal sub-manifold T G . the determinant of the metric of the six-dimensional base (cid:112) g ( y (cid:48) ) in (3.19) scales as g s ,whereas here it scales as g s ; two, the M p scaling in (3.19) is different from what we havehere; three, there is an extra factor of z n α in (3.50) that is absent in (3.19); and four, theappearance of the volume factors V and ˆ V compared to their absence in (3.19). Thevolume factor ˆ V is important, and is related to the effective volume of the base sub-manifold M . Its presence here signifies the fact that we have integrated over the fullinternal space (2.3), including the contributions from the derivatives along M in (3.2).On the other hand, V is extracted from an integral (cid:82) d y α g ( y α ) √ g ≈ g ( y α ) V and is the unwarped volume of M .The factor of g s that appears in (3.50) for n = 2 matches up with the similar de-pendence in (3.19), but now there is more: the generic dependence becomes g n / s ,implying that other powers of inverse g s should appear both in the sum as well as in theexponent. This differs from (3.19) where the factor of g s is universal over the whole rangeof ( { l i } , n i ). For large enough n , it appears that the exponential term is heavily sup-pressed killing the contributions to the energy-momentum tensor altogether as seen fromthe individual components of the energy-momentum tensor: T (np;3) ab ( z ) = (cid:88) k c k (cid:34) g / s + (cid:88) n k V g n ) / s (cid:16) g θ +4 / s (cid:17)(cid:35) exp (cid:32) − (cid:88) n k V g n ) / s · g θs (cid:33) (3.51) T (np;3) µν ( z ) = (cid:88) k c k (cid:34) g − / s + (cid:88) n k V g n ) / s (cid:16) g θ − / s (cid:17)(cid:35) exp (cid:32) − (cid:88) n k V g n ) / s · g θs (cid:33) T (np;3) mn ( z ) = T (np;3) αβ ( z ) = (cid:88) k c k (cid:34) g − / s + (cid:88) n k V g n ) / s (cid:16) g θ − / s (cid:17)(cid:35) exp (cid:32) − (cid:88) n k V g n ) / s · g θs (cid:33) . – 70 –owever the scaling analysis shows that θ = (2 + n ), so the series could be summed asin (3.23) instead. Putting a bound like (3.24), one could even get a convergent series (seefootnote 50). Any remaining divergence can be controlled by F (1) ( x − z ), F (1) ( z − z (cid:48) ) and F (1) ( x − z (cid:48) ) as before, but now there is also the volume factors, although new subtletiesappear because of the n factor in (3.50) and (3.51). Before we discuss how to control this,let us express the generic form of the energy-momentum tensor from (3.51) in the followingway: T (np;3)M a N a ( z ) = (cid:88) k c k (cid:34) g l a s + (cid:88) n k V g n ) / s (cid:16) g θ + l a s (cid:17)(cid:35) exp (cid:32) − (cid:88) n k V g n ) / s · g θs (cid:33) , (3.52)where the g s dependence in the exponential part for θ = (2 + n ) go as g / s , as before.The M p dependence is now useful to quantify. If the derivative action do not act on (3.48),most of the changes in red in (3.50) do not appear, and the energy-momentum tensor takesthe following form: T (np;4)MN ( z ) = (cid:88) { l i } ,n i ,k c k g MN ( z ) exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g / s (cid:32) F (1) ( z − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) ˆ V (cid:35) + (cid:88) { l i } ,n i ,k kc k V M σ ( { l i } ,n i ) − p (cid:90) d x F (1) ( x − z ) g / s · δ (cid:16)(cid:112) g ( z ) Q ( { l i } ,n i )T ( z ) (cid:17) δ g MN ( z ) (cid:115) g ( x ) g ( z ) × exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g / s (cid:32) F (1) ( x − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) ˆ V (cid:35) , (3.53) whose behavior is very similar to what we had in (3.19). For example when θ = 4 /
3, onemay easily see that the exponential factor goes as exp (cid:16) − k ˆ V g / s (cid:17) , with the integrand onthe second term behaving similar to (3.19). This series can be controlled, as we discussedin much details earlier (see footnote 50). The interesting question then is the scenariowhere the derivatives act on (3.48). Taking one derivative on (3.48), brings down a factorof − y α M p , but there is also a factor of inverse M p from (3.10), so that the overall scaling is − y α M p . This is good, but the worrisome feature is the minus sign, which will occur everytime we take an odd number of derivatives. Fortunately this can be cured from the startwhen we sum the trans-series in (3.16). Recall that the aim of such a summation process isto convert any series with terms like ± f ( z ) g s to exp (cid:16) − f ( z ) g s (cid:17) , so that when g s →
0, or f ( z ) →∞ , the exponential factor becomes very small, irrespective of the sign of the individualterms in the series. To deal with this, let us divide the n derivatives as n = ( n Ω , n Q )where n Ω is the number of derivatives that act on (3.48), or alternatively on G MN ab in(3.2), and n Q is the number of derivatives that act on everything else in (3.2). We can nowinsert a factor of ( − n Ω in the exponential piece (3.18). This way, once we take θ = 4 / p as M n Ω p exp (cid:16) − M n Ω p g / s (cid:17) thatbecomes smaller as n increases Similar story could be developed for other values of θ > A more acute question is what happens when the derivative action removes the M p factor. This happens,for example when the first derivative action brings down a − y α M p factor, and the second derivative action – 71 –nd show convergences there. We could also go back to (3.19), and insert two changes:one, use (3.48) and two, insert the warped volume of T G . We will however continue to keepall functional dependences on M × M for simplicity. The result is: T (np;5)MN ( z ) = (cid:88) { l i } ,n i ,k c k g MN ( z ) exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g − n ) / s (cid:32) F (1) ( z − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) V T (cid:35) + (cid:88) { l i } ,n i ,k kc k M n p V T M σ ( { l i } ,n i ) − p (cid:90) d x (2 z α ) n F (1) ( x − z ) g n ) / s · δ (cid:16)(cid:112) g ( z ) Q ( { l i } ,n i )T ( z ) (cid:17) δ g MN ( z ) (cid:115) g ( x ) g ( z ) × exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g − n ) / s (cid:32) F (1) ( x − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33) V T (cid:35) , (3.54) where g is the warped determinant of the metric for the base M × M , whereas g is the un-warped determinant. V T is the un-warped volume of the orthogonal space T G .Interestingly, the g s and the M p behavior of this is similar to (3.50), the KKLT instanton,and because of that, the series (3.54) should be convergent (in the sense discussed infootnote 50). Interestingly, once we remove the volume dependence, the result becomes: T (np;6)MN ( z ) = (cid:88) { l i } ,n i ,k c k g MN ( z ) exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g n / s (cid:32) F (1) ( z − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33)(cid:35) + (cid:88) { l i } ,n i ,k kc k M n p M σ ( { l i } ,n i ) − p (cid:90) d x (2 z α ) n F (1) ( x − z ) g n ) / s · δ (cid:16)(cid:112) g ( z ) Q ( { l i } ,n i )T ( z ) (cid:17) δ g MN ( z ) (cid:115) g ( x ) g ( z ) × exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g n / s (cid:32) F (1) ( x − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33)(cid:35) . (3.55) Let us summarize what we have so far. Assuming no dependency on the coordinates of T G there are two possible choices for Ω ab given in (3.48): one with B = 0 and the other withnon-zero B . For both cases, the g s scalings of the energy-momentum tensors for the BBSand the KKLT instanton gases behave differently depending on whether the orthogonalvolumes of the internal sub-manifold are taken into account or not. For the BBS instantongas, the energy-momentum tensor without switching on the volume of T G is given in (3.19).Once we switch on the volume of T G , the result changes to (3.54), with n = n Ω , where n Ω is the number of derivatives acting on (3.48). On the other hand, for the KKLT instantongases, once we switch on the volume of the sub-manifold M , the energy-momentum tensorbecome (3.50). If not, the result is: removes y α . In fact the combined action also removes the M p factor. However the exponential piece doesretain the information of the derivative action that acts on the exponential factor as well as the otherderivative action. This means, no matter how the derivatives act, there would always be a factor that goesto zero as: lim n Ω →∞ M n Ω p exp (cid:16) − M n Ω p g / s (cid:17) → , which for M p → ∞ goes to zero even faster. The conclusion remains unchanged for g s <
1. For g s → p to go to infinity faster than some given power of g s . For example imagine theexponential part is exp (cid:0) − M k p g k s (cid:1) , and let g s goes to zero as g s → (cid:15) . Then as long as M p goes to infinityas M p → (cid:15) − κ with κ > k k , the exponential part vanishes. This way convergence can be attained. – 72 – (np;7)MN ( z ) = (cid:88) { l i } ,n i ,k c k g MN ( z ) exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g n / s (cid:32) F (1) ( z − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33)(cid:35) + (cid:88) { l i } ,n i ,k kc k M n p M σ ( { l i } ,n i ) − p (cid:90) d x ( − z α ) n F (1) ( x − z ) g n / s · δ (cid:16)(cid:112) g ( z ) Q ( { l i } ,n i )T ( z ) (cid:17) δ g MN ( z ) (cid:115) g ( x ) g ( z ) × exp (cid:34) − k (cid:90) d z (cid:48) (cid:112) g ( z (cid:48) ) g n / s (cid:32) F (1) ( x − z (cid:48) ) Q ( { l i } ,n i )T ( z (cid:48) )M σ ( { l i } ,n i ) − p (cid:33)(cid:35) , (3.56) which has a different g s scaling as expected. Once n ≡ n Ω = 0, we are basically dealingwith B = 0 in (3.48). The results for all the cases are summarized in Tables 3 and .Before ending this section let us resolve couple other issues that we have been draggingalong with our formulae so far. The first one is the appearance of an integral of the form: (cid:90) d x (cid:112) − g ( x ) F (1) ( x − z ) exp (cid:20) − M n Ω p g / s (cid:90) d y (cid:48) F (1) ( x − y (cid:48) ) A e ( y (cid:48) ) (cid:21) ≡ g − / s G e ( z, g s , n ) , (3.57) where A ( y (cid:48) ) contains the quantum series (3.2) etc., and n Ω = n . The integral over d x is finite and well-defined, much like what we had in (3.22). The extra exponential pieceonly makes the convergence better, both in terms of the integral and in terms of the choice n >>
1, with the function G e ( z, g s , n ) vanishing for the latter case. The smoothness ofthe warp-factor H ( y ) and the localized nature of the non-locality function F (1) ( x − y (cid:48) ) arealso the factors contributing to the convergence. On the other hand, the integral over d y (cid:48) is well-defined because of the compactness of the internal manifold M × M .The second issue has to do with the appearance of the metric components in the non-perturbative energy-momentum tensors T (np; l )MN , i.e. the metric factor g MN ( z ) accompanying(3.19), (3.50), (3.53), (3.54), (3.55) and (3.56). They typically blow-up in the limit g s → f ( x ) ≡ (cid:88) n ≥ d n x n = d x + d x + d x + d x + ........d n ≡ (cid:88) l ≥ c l ( − l ) n n ! = c ( − n n ! + c ( − n n ! + c ( − n n ! + c ( − n n ! + ......., (3.58)with ( d i , c j ) to be arbitrary constants. Once the set of d i are specified, the corresponding c j could be easily determined by inverting a certain matrix whose details appear in footnote48. We can now combine things together to express the following series: xd = c ( − x + c ( − x + c ( − x + c ( − x + ......x d = c ( − x + c ( − x + c ( − x + c ( − x + ......x d = c ( − x + c ( − x + c ( − x + c ( − x + ......, (3.59)and so on. The point of this obvious exercise was to justify one little thing, which becomesapparent when we add every term vertically down. Summing vertically down we easily get:– 73 –nstanton type V a V b T g s scaling T (np)MN θ min BBS a = b = 0 g θ + las g n / s (3.55) ( n + 4)Delocalized BBS a = 0 , b = 1 g θ + las g n / s (3.54) ( n + 2)KKLT a = b = 0 g θ + las g n / s (3.56) ( n + 1)Delocalized KKLT a = 1 , b = 0 g θ + las g n / s (3.50) ( n + 2) Table 4:
The g s scalings of the four kind of wrapped M5-instantons that contribute to the non-perturbative energy-momentum tensor T (np)MN , but now with the two-form Ω as defined to be (3.48).Again, the g s and M p scalings remain unchanged for the two cases with B = 0 and B (cid:54) = 0, ifthe derivatives do not act on (3.48). The other parameter θ , θ min and l a are defined as before. f ( x ) = c (cid:0) e − x − (cid:1) + c (cid:0) e − x − (cid:1) + c (cid:0) e − x − (cid:1) + c (cid:0) e − x − (cid:1) + ... = (cid:88) k ≥ c k (cid:16) e − kx − (cid:17) , (3.60) which is all we need. The above result remains unchanged no matter what c i we choose,positive or negative. The issue of convergence of such a series (both in terms of k and( { l i } , n i )) has already been dealt with earlier, so we will not discuss it further here. If wenow identify f ( x ) with, say, U (1)( { l i } ,n i ) ( y ) in (3.16) and x by the integral structure therein,then the action S in (3.29), gets modified to the following indefinite integral structure: S (cid:48) = (cid:88) { l i } ,n i ,k (cid:90) d x √− g ∞ (cid:88) r =1 c k (cid:20) exp (cid:16) − k M p (cid:90) d y (cid:112) g ( y ) F ( r ) ( x − y ) W ( r − ( y ; { l i } , n i ) (cid:17) − (cid:21) , (3.61) in precisely the same way as in our simple exercise (3.60). In fact all the non-perturbativeactions that we wrote as a trans-series should be modified in the aforementioned way.This addition of a counter-term with a relative minus sign in (3.61), and subsequently inall other actions, removes the extra metric dependences from all the energy-momentumtensors, because: lim g s → (cid:104) g MN exp (cid:16) − kg | θ | s (cid:17) − g MN (cid:105) → , (3.62)for all θ >
0, with similar cancellations for every k . To summarize then, the g s scalings ofall the non-perturbative energy-momentum tensors are exactly as they appear in Tables3 and without any superfluous metric factors. With all the perturbative and non-perturbative quantum corrections at hand, includingbeing equipped with the construction of the Glauber-Sudarshan states, we are ready tostudy the basics EOMs governing them. Some parts of the EOMs have already been– 74 –iscussed in details in [21], so we will not take that path here. Instead we will analyze theEOMs using the Schwinger-Dyson equations [38]. The full M-theory action, that includesall the corrections that we studied in section 3.1 and 3.2, can be written as: S tot ≡ S + S (cid:48) + S + S + ... = S + S np + S b + S top , (3.63)where S is the M-theory action in (3.1) that includes the infinite collection of perturbativelocal and non-local corrections, including the action for M2 and fractional M2 branes; S np is the action for the instanton gas studies in sections 3.2.1 and 3.2.3 that includes S (cid:48) from(3.61) and other contributions that we elaborated in section 3.2.3; S top is the topologicalpart of M-theory action that we studied in full details in [21], so we don’t discuss it here;and S b is the action of the branes and surfaces, including their fermionic and higher orderinteractions, that do not appear in S . As an example, we have uplifted-six-brane fermionicand higher order interactions in S and S given as (3.30) and (3.41) respectively. Theother wrapped brane actions may be thought of as coming from the non-local interactionsand topological action that we discussed in S and S top from (3.1) and [21] respectively.Note also that each of the pieces in (3.63) has an infinite number of terms, so S tot is prettymuch an exhaustive collection. S tot is also in some sense a Wilsonian action constructed by integrating out the UVmodes in the solitonic background. As we elaborated earlier, such integrating out pro-cedure is possible because our vacuum is supersymmetric and solitonic, so do not sufferfrom any pathologies attributed to vacua like Bunch-Davies and other similar avatars. Su-persymmetry is broken spontaneously from switching on coherent states of non-self-dualG-fluxes as in (2.95), so the positive cosmological constant Λ appears from a conspiracybetween these fluxes and quantum corrections in a way discussed in [21] with the zero pointenergy playing no part here. This is of course the advantage we get from our choice ofvacuum, but here we want to inquire about the stability of the Glauber-Sudarshan stateamidst the infinite set of quantum corrections emanating from S tot . This is where theSchwinger-Dyson equations [38] become immensely useful.The original formulation of the Schwinger-Dyson equations (SDEs) in [38] providerelations between Green’s functions in QFT as expectation values over states. In thegravitational and the flux sectors the SDEs imply: (cid:90) [ D g MN ] δδg PQ ≡ ≡ (cid:90) [ D C MNP ] δδ C PQR , (3.64) i.e. the integrals over total derivatives vanish, so it shouldn’t be too hard to get them here.However before we get the required SDEs, recall that in [21, 22] we carefully distinguishedbetween warped ( i.e. g s dependent) and un-warped ( i.e. g s independent) parts of themetric. Question is, in the computation (2.79), what metric was used? The answer issimple: for us there is only the solitonic background (2.1) with the modes Ψ k in (2.7) for thegravitational sector and Υ k (that we discuss below) for the flux sector . Everything else, in Not to be confused with the fermions Ψ and Υ used in section 3.2.2! – 75 –articular the background (2.4) and the corresponding time-dependent G-flux components,must appear as expectation values over the generalized Glauber-Sudarshan states .Looking at the integral form of the result on the RHS of (2.79), we see that α ( ψ ) µν ( k , t )appears as the expectation value. Since α ( ψ ) µν ( k , t ) captures the complete time-dependenceof the corresponding metric components, our analysis in (2.79) has resulted in a fully warped metric components from the path integral. In retrospect, this is what should have been, sothe apparent consistency is not much of a surprise, although one concern could be raisedhere: the O (cid:16) g as M bp (cid:17) corrections accompanying the expected answer. Does that mean wedeviate from the de Sitter background? The answer, as we shall soon see, is no: once wemake the right choice of the displacement operator, these correction terms do not appearanymore. What is interesting however is that this choice also paves the way to achieve thefully warped metric components from the SDEs, as will be elaborated below.There is yet another thing that needs elaboration before we proceed further, and has todo with the displacement operators D ( α ) defined in (2.61). The action of the displacementoperator is (2.57), but it hides the fact that there are both gravitational and four-formfields participating in the construction. This means the form of D ( α ) cannot be as simpleas (2.76), and the modified form should incorporate all the existing field components thatform the generalized Glauber-Sudarshan states in our set-up. This is a tedious exercise,but we can make it simple by resorting to few definitions. Let { α MN } denotes the set ofFourier components in (2.6); and { β MNP } denotes the corresponding set for the three-formflux components C MNP , then D ( α ) from (2.76) may be modified to: D ( α, β ) = exp (cid:20)(cid:90) + ∞−∞ d k (cid:16) α (Ψ)MN ( k , k ) (cid:101) g ∗ MN ( k , k ) + β (Υ)MNP ( k , k ) (cid:101) C ∗ MNP ( k , k ) (cid:17)(cid:21) (3.65) × exp (cid:20) − (cid:90) + ∞−∞ d k (cid:16) α (Ψ)MN ( k , k ) α ∗ (Ψ)MN ( k , k ) + β (Υ)MNP ( k , k ) β ∗ (Υ)MNP ( k , k ) (cid:17) + .... (cid:21) , where the tilde denote Fourier transforms, and the dotted terms are the higher order mixingbetween the various components coming from our generic definition of a † eff in (2.59). Thisis arranged in a way that (3.65) is not unitary. The other components in (3.65) are Ψ k andΥ k which are respectively the set of Schr¨odinger wave-functions in (2.7) and a similar setfor the three-form flux components C MNP . Note that C MNP are not gauge invariants, andneither are the metric components, so D ( α, β ) would change the expectation values of thecorresponding fields in the right way under gauge transformations. We also expect: (cid:104) α, β | α, β (cid:105) = (cid:104) Ω | D † ( α, β ) D ( α, β ) | Ω (cid:105) = (cid:82) [ D{ g MN } ] [ D{ C PQR } ] e i S tot D † ( α, β ) D ( α, β ) (cid:82) [ D{ g MN } ] [ D{ C PQR } ] e i S tot , (3.66)where using some abuse of notation, we have taken D ( α, β ) to denote both the operatorand the field. Which is which should be clear from the context. The set { g MN } denotesthe set of metric components ( g mn , g αβ , g ab , g µν ) and the set { C MNP } denotes the C-fields Recall from section 2.4 that we call the shifted interacting vacuum D ( σ ) | Ω (cid:105) as the generalized Glauber-Sudarshan states to distinguish it from the original Glauber-Sudarshan states created out of the shiftedharmonic vacuum D ( σ ) | (cid:105) . – 76 –hat appear from the G-flux components (G mnpq , G mnpα , G mnαβ , G mnpa , G mnab , G ijm ) andother permutations. These will be related to the components that we encountered insections 3.1 and 3.2, and also in [21, 22]. Finally, S tot is the fully interacting action writtenin (3.63) and is responsible for creating the interacting vacuum | Ω (cid:105) .There are still couple more issues that we need to clarify before we proceed further.First, the way we have expressed | α, β (cid:105) , it is clearly not normalized because D ( α, β ) is notunitary. The shifted vacuum | α, β (cid:105) ≡ D ( α, β ) | Ω (cid:105) , as we showed earlier using the simplerversion (2.74), does produce the expected answer in (2.79), so we expect the same to holdfor (cid:104){ g MN }(cid:105) ( α,β ) . This is a straightforward exercise so we will not do it here, instead wewant to point out that the expectation values of the G-flux components (cid:104){ G MNPQ }(cid:105) ( α,β ) now reproduce the expected results from [21, 22].Secondly, as we cautioned earlier, because of the presence of metric and C-fields, whichare not gauge invariant quantities, there should be Faddeev-Popov ghosts changing: S tot → S tot − S ghost , (3.67)where the relative sign is chosen for later convenience. If (3.67) is always true, then itwill make the action even more complicated than (3.63). There are specific gauge choicesthat do not create propagating ghosts, when we take the metric and the C-field degreesof freedom separately, but it is not clear such a gauge exists when we take everything together . This would mean that we might have to venture beyond (3.63). Taking all theseinto considerations, the first set of Schwinger-Dyson equations resulting from (3.66) thentakes the following form: (cid:28) δ S tot δ { g MN } (cid:29) ( α,β ) = (cid:28) δ S ghost δ { g MN } (cid:29) ( α,β ) − (cid:28) δδ { g MN } log (cid:16) D † ( α, β ) D ( α, β ) (cid:17)(cid:29) ( α,β ) (3.68) (cid:28) δ S tot δ { C MNP } (cid:29) ( α,β ) = (cid:28) δ S ghost δ { C MNP } (cid:29) ( α,β ) − (cid:28) δδ { C MNP } log (cid:16) D † ( α, β ) D ( α, β ) (cid:17)(cid:29) ( α,β ) , where all degrees of freedom appear on both sides of the two set of equations, making it acomplicated set of coupled differential equations. Question is how to solve these equationsto extract useful data for the generalized Glauber-Sudarshan states.First, even without solving anything we see that the SDEs’ are expressed as expectationvalues over the generalized Glauber-Sudarshan states. This is already a good start asour earlier path integral approach in (2.79) showed us that the expectation value of themetric over the state | α (cid:105) does reproduce the de Sitter space-time. Secondly, the functionalderivatives are taken with respect to the space-time metric and C-field components so itwould be useful to bring (3.65) to the space-time integral format. This becomes: D ( α, β ) = exp (cid:20)(cid:90) + ∞−∞ d x √− g (cid:16) α (Ψ)MN ( x ) g MN ( x ) + β (Υ)MNP ( x ) C MNP ( x ) (cid:17)(cid:21) × exp (cid:20) − (cid:90) + ∞−∞ d x √− g (cid:18)(cid:12)(cid:12)(cid:12) α (Ψ)MN ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) β (Υ)MNP ( x ) (cid:12)(cid:12)(cid:12) (cid:19) + ...... (cid:21) , (3.69)where we have used the normalization condition from footnote 5 to bring the second line inthe right form. The metric determinant is of the solitonic background (2.1) which means– 77 – − g = h − h / . Plugging (3.69) in (3.68) reproduces: δδ { C MNP ( z ) } (cid:104) log (cid:16) D † ( α, β ) D ( α, β ) (cid:17)(cid:105) = 2 h − ( z ) h / ( z ) β (Υ)MNP ( z ) + .... (3.70) δδ { g MN ( z ) } (cid:104) log (cid:16) D † ( α, β ) D ( α, β ) (cid:17)(cid:105) = (cid:16) − g M (cid:48) N (cid:48) ( z ) g M (cid:48) N (cid:48) ( z ) (cid:17) h − ( z ) h / ( z ) α (Ψ)MN ( z ) + .., where the dotted terms are higher order mixing terms resulting from (2.59). Note that theresults, at least to the order that we study here, do not depend directly on D ( α, β ), but areproportional to the background metric (2.4) and the corresponding G-flux components. Itis also interesting to ask how does the expectation value, computed in (2.78), change if twodifferent generalized Glauber-Sudarshan states are used, for example | α , β (cid:105) and | α , β (cid:105) . Ifwe ignore β for simplicity, then we ask how the expectation value (cid:104) α | g µν | α (cid:105) differs from(2.78). In path integral form, the numerator takes the following form: (cid:90) [ D g µν ] e iS D † ( α ) δg µν D ( α ) = (cid:32)(cid:89) k (cid:90) d ( Re (cid:101) g µν ( k )) d ( Im (cid:101) g µν ( k )) (cid:33) exp (cid:34) iV (cid:88) k k | (cid:101) g µν ( k ) | + iS sol + .. (cid:35) × exp (cid:40) V (cid:88) k (cid:48) (cid:104) Re (cid:16) α ( ψ )(1) µν ( k (cid:48) ) + α ( ψ )(2) µν ( k (cid:48) ) (cid:17) − i Im (cid:16) α ( ψ )(2) µν ( k (cid:48) ) − α ( ψ )(1) µν ( k (cid:48) ) (cid:17)(cid:105) Re (cid:101) g µν ( k (cid:48) ) (cid:41) × exp (cid:40) V (cid:88) k (cid:48) (cid:104) Im (cid:16) α ( ψ )(1) µν ( k (cid:48) ) + α ( ψ )(2) µν ( k (cid:48) ) (cid:17) + i Re (cid:16) α ( ψ )(2) µν ( k (cid:48) ) − α ( ψ )(1) µν ( k (cid:48) ) (cid:17)(cid:105) Im (cid:101) g µν ( k (cid:48) ) (cid:41) × V (cid:88) k (cid:48)(cid:48) ψ k (cid:48)(cid:48) ( x , y, z ) e − ik (cid:48)(cid:48) t (cid:0) Re (cid:101) g µν ( k (cid:48)(cid:48) ) + i Im (cid:101) g µν ( k (cid:48)(cid:48) ) (cid:1) exp (cid:32) − V (cid:88) k (cid:48) | α ( ψ ) µν ( k (cid:48) ) | (cid:33) , (3.71) where δg µν implies we have ignored the solitonic part of the field g µν . We see thatthe coefficients of Re (cid:101) g µν ( k (cid:48) ) and Im (cid:101) g µν ( k (cid:48) ) become complex . Interestingly the complexfactor is α ( ψ )(2) µν ( k (cid:48) ) − α ( ψ )(1) µν ( k (cid:48) ) and therefore vanishes when α = α . The denominator willhave a similar form as (3.71) except without the metric field. The integral can be easilyperformed, and here for illustrative purpose let us assume that (cid:101) g µν ( k (cid:48) ) is real. Puttingeverything together we get : (cid:104) α | g µν | α (cid:105) = η µν h / ( y, x ) + 12 (cid:90) d k ω ( ψ ) k ˆ α ( ψ ) µν ( k , t ) ψ k ( x , y, z ) + O (cid:18) g cs M dp (cid:19) (3.72)ˆ α ( ψ ) µν ( k , t ) ≡ Re (cid:16) α ( ψ )(1) µν ( k , t ) + α ( ψ )(2) µν ( k , t ) (cid:17) − i Im (cid:16) α ( ψ )(2) µν ( k , t ) − α ( ψ )(1) µν ( k , t ) (cid:17) , which tells us that unless α ( ψ )(1) µν ( k , t ) = α ( ψ )(2) µν ( k , t ) = α ( ψ ) µν ( k , t ) where α ( ψ ) µν ( k , t ) is thevalue from (2.17), the expectation value cannot produce a de Sitter space. Additionally,the inequality between α ( ψ )( i ) µν ( k , t ) suggests that (3.72) may not even be real. This means,an equality between α ( ψ )( i ) µν ( k , t ) only guaranties a de Sitter space when α ( i )( ψ ) µν ( k , t ) takesthe value in (2.17), otherwise it will be another time-dependent space-time. In this section both fields and operators of the solitonic background (2.1) will be denoted by Romanletters i.e. g MN and C MNP , whereas the fields of the background (2.4) will be denoted by bold-faced letters i.e. g MN and C MNP . In this way connecting to variables from sections 3.1 and 3.2 will be easier. – 78 –uch a criterion is particularly useful when we evaluate the expectation values of theproducts of metric and G-flux components. A simple example would be the expectationvalue (cid:104) α, β | g µν ( z ) g µν ( z ) | α, β (cid:105) , where |{ α }(cid:105) ≡ | α (cid:105) denotes the coherent states associatedwith the metric sector and |{ β }(cid:105) ≡ | β (cid:105) denotes the coherent states associated with theG-flux sector. In the mixed sector, as we discussed earlier, the coherent states may bedenoted as | α, β (cid:105) . For the simple case, once we concentrate on the gravitational sector, weexpect the following decomposition: (cid:104) g µν ( z ) g µν ( z ) (cid:105) α ≡ (cid:104) α | g µν ( z ) g µν ( z ) | α (cid:105) = (cid:90) d α (cid:48) π (cid:104) α | g µν ( z ) | α (cid:48) (cid:105)(cid:104) α (cid:48) | g µν ( z ) | α (cid:105) , (3.73)where we have imposed the completeness property of the Glauber-Sudarshan states. Here | α (cid:105) is related to α ( ψ ) µν ( k , t ) from (2.17), but | α (cid:48) (cid:105) could in principle be arbitrary. This means (cid:104) α (cid:48) | g µν ( z ) | α (cid:105) from (3.72) isn’t necessarily a de Sitter space, unless α (cid:48) = α . Thus thedecomposition (3.73) implies: (cid:104)| g µν ( z ) | (cid:105) α ≡ (cid:104) g µν ( z ) g µν ( z ) (cid:105) α = |(cid:104) g µν ( z ) (cid:105) α | + c (cid:88) α (cid:48) (cid:54) = α |(cid:104) α | g µν ( z ) | α (cid:48) (cid:105)| (3.74)where the sum is over backgrounds of the form (3.72), with α = α and α = α (cid:48) , thatdeviate from the de Sitter space; and c is a constant that is required to convert the integralin (3.73) to a sum. In a similar vein, any powers of metric or G-flux components, or evenmixed powers of metric and flux components would have at least a decomposition of theform (3.74). The sum in (3.74) involve terms with α (cid:48) > α as well as with α (cid:48) < α . Theycome with opposite signs in (3.72), so it will be worthwhile to evaluate this directly fromthe path integral. The integral that we are looking for now is: (cid:90) [ D g µν ] e iS D † ( α ) | g µν | D ( α ) = (cid:32)(cid:89) k (cid:90) d ( Re (cid:101) g µν ( k )) d ( Im (cid:101) g µν ( k )) (cid:33) exp (cid:34) iV (cid:88) k k | (cid:101) g µν ( k ) | + iS sol + ... (cid:35) × V exp (cid:34) V (cid:88) k (cid:48) (cid:16) Re α ( ψ ) µν ( k (cid:48) ) Re (cid:101) g µν ( k (cid:48) ) + Im α ( ψ ) µν ( k (cid:48) ) Im (cid:101) g µν ( k (cid:48) ) (cid:17) + ... (cid:35) exp (cid:32) − V (cid:88) k (cid:48) | α ( ψ ) µν ( k (cid:48) ) | (cid:33) × (cid:88) k (cid:48)(cid:48) ,k (cid:48)(cid:48)(cid:48) ψ k (cid:48)(cid:48) ( x , y, z ) ψ k (cid:48)(cid:48)(cid:48) ( x (cid:48) , y (cid:48) , z (cid:48) ) e − i ( k (cid:48)(cid:48) t + k (cid:48)(cid:48)(cid:48) t (cid:48) ) (cid:16) Re (cid:101) g µν ( k (cid:48)(cid:48) ) + i Im (cid:101) g µν ( k (cid:48)(cid:48) ) (cid:17)(cid:16) Re (cid:101) g µν ( k (cid:48)(cid:48)(cid:48) ) + i Im (cid:101) g µν ( k (cid:48)(cid:48)(cid:48) ) (cid:17) , (3.75) which is basically the numerator of the expectation value (3.73), except that we have ig-nored the solitonic part of the metric. This can be easily rectified. Note that we have sep-arated the two metric components over space and time so that short distance singularitiesmay be avoided. we will also avoid summing over repeated indices to avoid overcomplicat-ing the integral. This means the tensorial property of the metric is not much of a concernhere, and assuming this to be the case, the integral (3.75) may be evaluated with the aidof a few notations. Let ( k, k (cid:48) , k (cid:48)(cid:48) , k (cid:48)(cid:48)(cid:48) ) ≡ ( k p , k n , k m , k l ) and (cid:101) g µν ( k ) ≡ Φ( k p ) ≡ Φ p . We willassume Re α ( ψ ) µν ( k (cid:48) ) = Im α ( ψ ) µν ( k (cid:48) ) ≡ α n for simplicity that can be easily relaxed. As willbe clear, none of these assumptions are necessary, and more importantly do not effect thefinal conclusion, so over-complicating the analysis will lead to the same conclusion as with– 79 –he simpler version that we choose here. As an exercise, the reader could verify this indetails. We will also go to the Euclidean formalism so that ik → − k ≡ − k p . With thesechanges, the integral (3.75), now becomes: Num (cid:2) (cid:104)| g µν | (cid:105) α (cid:3) = (cid:32)(cid:89) p (cid:90) d ( Re Φ p ) d ( Im Φ p ) (cid:33) exp (cid:34) − V (cid:88) p k p (cid:0) ( Re Φ p ) + ( Im Φ p ) (cid:1) + ... (cid:35) × exp (cid:34) V (cid:88) n α n ( Re Φ n + Im Φ n ) + ... (cid:35) exp (cid:32) − V (cid:88) n | α n | (cid:33) × V (cid:88) m,l ψ m ( x, y, z ) ψ l ( x (cid:48) , y (cid:48) , z (cid:48) ) (cid:16) Re Φ m + i Im Φ m (cid:17)(cid:16) Re Φ l + i Im Φ l (cid:17) , (3.76) where ψ m ( x, y, z ) ≡ ψ k (cid:48)(cid:48) ( x , y, z ) e − ik (cid:48)(cid:48) t . The form of the integral is somewhat similar tothe integral one would encounter when computing the two-point function. However thereis a crucial difference: the presence of exp (cid:2) V (cid:80) n α n ( Re Φ n + Im Φ n ) (cid:3) factor that wasresponsible in (2.71), (2.78) and (2.82) to give non-zero results for one-point functions by shifting the vacuum. Here, and because of this, the integral in (3.76) cannot just be theresult that we know for the two-point function. There will be more contributions thattypically vanish in the usual computation of the two-point function. To quantify this, letus express the contributions from the last line of (3.76) as four sectors: (+ +) : Re Φ m Re Φ l − Im Φ m Im Φ l + i ( Im Φ m Re Φ l + Re Φ m Im Φ l )(+ − ) : Re Φ m Re Φ l + Im Φ m Im Φ l + i ( Im Φ m Re Φ l − Re Φ m Im Φ l )( − +) : Re Φ m Re Φ l + Im Φ m Im Φ l − i ( Im Φ m Re Φ l − Re Φ m Im Φ l )( − − ) : Re Φ m Re Φ l − Im Φ m Im Φ l − i ( Im Φ m Re Φ l + Re Φ m Im Φ l ) , (3.77) where we have used Φ( − k m ) ≡ Φ − m = Φ ∗ m . In the usual computation in QFT, all theimaginary pieces ( i.e. the coefficients of i and not Im Φ m ) in (3.77) cancel out because theyform linear terms in a gaussian integral. Clearly this cannot be the case now. Similarly, allthe real pieces in the (+ +) sector also cancel out because of the relative minus sign. Thebehavior of the ( − +) and ( − − ) sector would be similar to the (+ +) and (+ − ) sector sowe could concentrate only on the first two. However for each of the two sectors we couldeither have m (cid:54) = l or m = l . The result of the integrals for each of the sector then yieldsthe following:(+ +) , ( m = l ) : i (cid:32)(cid:89) p k p (cid:33) (cid:88) m α m k m ψ m ( x, y, z ) ψ m ( x (cid:48) , y (cid:48) , z (cid:48) ) (3.78)(+ +) , ( m (cid:54) = l ) : i (cid:32)(cid:89) p k p (cid:33) (cid:88) m,l α m α l k m k l ψ m ( x, y, z ) ψ l ( x (cid:48) , y (cid:48) , z (cid:48) )(+ − ) , ( m (cid:54) = l ) : 12 (cid:32)(cid:89) p k p (cid:33) (cid:88) m,l α m α l k m k l ψ m ( x, y, z ) ψ ∗ l ( x (cid:48) , y (cid:48) , z (cid:48) )(+ − ) , ( m = l ) : 12 (cid:32)(cid:89) p k p (cid:33) (cid:88) m (cid:18) α m k m − k m (cid:19) ψ m ( x, y, z ) ψ ∗ m ( x (cid:48) , y (cid:48) , z (cid:48) ) , – 80 –here one may easily verify that, when α p = 0, all the contributions vanish, except for oneterm from the (+ − ) sector with m = l . In fact this term is exactly the propagator for thegravitons as may be seen from the following computation: (cid:88) m ψ m ( x, y, z ) ψ ∗ m ( x (cid:48) , y (cid:48) , z (cid:48) ) k m = (cid:90) d k ψ k ( x , y, z ) ψ ∗ k ( x (cid:48) , y (cid:48) , z (cid:48) ) e − ik ( t − t (cid:48) ) k + i(cid:15) , (3.79)where the infinite product coefficient in (3.78) is cancelled by the denominator of the pathintegral. The spatial wave-function ψ k appears from (2.7) and contributes to (2.6), withthe i(cid:15) factor taking care of the residue at the poles in the usual way.The results in (3.78), confirms the generic expectation (3.74), and the first term whichis the square of the expectation value (cid:104) g µν (cid:105) α is basically the first term in the sector (+ − )for m = l . In the language of (cid:104) g µν (cid:105) α , the squaring would involve two integrals d kd k (cid:48) ,but ( k, k (cid:48) ) are related by δ ( k + k (cid:48) ) (recall we are in the (+ − ) sector hence k = − k (cid:48) and not k = k (cid:48) ), so we do get the first term from (+ − ) with m = l correctly. Note that α m = α m | i | , so the results fit well, confirming in turn the decomposition (3.74).Generalizing this, we expect, for example: (cid:28) R MN − g MN R (cid:29) ( α,β ) = (cid:104) R MN (cid:105) ( α,β ) − (cid:104) g MN (cid:105) ( α,β ) (cid:104) R (cid:105) ( α,β ) + ..... (3.80)= (cid:104) R MN (cid:105) ( α,β ) − (cid:104) g MN (cid:105) ( α,β ) (cid:104) g PQ (cid:105) ( α,β ) (cid:104) R PQ (cid:105) ( α,β ) + ....., where the dotted terms would be the sum of the terms where we allow intermediate stateslike ( α (cid:48) , β (cid:48) ) , ( α (cid:48)(cid:48) , β (cid:48)(cid:48) ) etc. to appear with the condition that α (cid:54) = ( α (cid:48) , α (cid:48)(cid:48) ) and β (cid:54) = ( β (cid:48) , β (cid:48)(cid:48) ).Again, since we are in the gravitational sector, we won’t need the information for β in(3.80), so we will suppress it. This means we can express the expectation value of the Riccicurvature in the following suggestive way: (cid:104) R MN (cid:105) α = − (cid:2) ∂ P ∂ Q (cid:104) g MN (cid:105) α + ∂ M ∂ N (cid:104) g PQ (cid:105) α − ∂ (M ∂ | P | (cid:104) g N)Q (cid:105) α (cid:3) (cid:104) g PQ (cid:105) α (3.81)+ 12 (cid:20) ∂ M (cid:104) g PQ (cid:105) α ∂ N (cid:104) g RS (cid:105) α + ∂ P (cid:104) g MQ (cid:105) α ∂ [R (cid:104) g | N | S] (cid:105) α (cid:21) (cid:104) g PR (cid:105) α (cid:104) g QS (cid:105) α − (cid:2) ∂ (M (cid:104) g N)Q (cid:105) α − ∂ Q (cid:104) g MN (cid:105) α (cid:3) [2 ∂ Q (cid:104) g RS (cid:105) α − ∂ S (cid:104) g PR (cid:105) α ] (cid:104) g PR (cid:105) α (cid:104) g QS (cid:105) α + .... where the symbol | P | stands for the index neutral to symmetrization or anti-symmetrization,and the dotted terms are the ones that have ( α (cid:48) , α (cid:48)(cid:48) , .. ) intermediate states with none ofthem equal to α . The results (3.81) and (3.80) convey something very important, once wenote that (cid:104) g MN (cid:105) α = g MN from (2.78), where g MN is precisely the warped , i.e. g s dependent,metric from [21, 22]. The two equations, (3.81) and (3.80), and especially (3.80), tell usthat the expectation value of the Einstein tensor over the generalized Glauber-Sudarshanstates has a part that is exactly the Einstein tensor computed using the metric (2.4)!The above conclusion is important so let us summarize what we have so far. Given asolitonic background (2.1) and the corresponding G-flux components to support it, we canconstruct generalized Glauber-Sudarshan states over it. Expectation values of the metric– 81 –nd the G-flux components on these states give us the time-dependent background (2.4),and the corresponding time-dependent G-flux components. Not only that, it now appearsthat the expectation values of the metric and the flux EOMs have parts that are preciselythe EOMs for the metric (2.4) and the corresponding G-flux components. In other words,we can quantify the above statements by first noting: (cid:28) δ S tot δ { g MN } (cid:29) σ = δ S ( σ )tot δ (cid:104) g MN (cid:105) σ + (cid:88) σ (cid:48) (cid:54) = σ (cid:28) δ S tot δ { g MN } (cid:29) ( σ | σ (cid:48) ) (cid:28) δ S tot δ { C MNP } (cid:29) σ = δ S ( σ )tot δ (cid:104) C MNP (cid:105) σ + (cid:88) σ (cid:48) (cid:54) = σ (cid:28) δ S tot δ { C MNP } (cid:29) ( σ | σ (cid:48) ) , (3.82)where S ( σ )tot ≡ S tot ( (cid:104) g PQ (cid:105) σ , (cid:104) C PQR (cid:105) σ ), which is basically the RHS of (3.1) defined in termsof the warped metric and flux components, g MN , C MNP and G MNPQ . The perturbativequantum terms will then be (3.2), and the non-perturbative terms will be as elaboratedin section 3.2. The other quantities are defined as follows: σ ≡ ( α, β ) and ( σ | σ (cid:48) ) denotethe intermediate generalized Glauber-Sudarshan states { σ (cid:48) } . As such the sum in (3.82) isover those states with the condition that they do not equal { α } , at least not all of them.Combining (3.82) with (3.68), then leads to the following set of equations: δ S ( σ )tot δ (cid:104) g MN (cid:105) σ = δ S ( σ )tot δ (cid:104) C MNP (cid:105) σ = 0 , (3.83)which are exactly the EOMs that we encountered, and solved, in [21] and [22]! Appear-ance of these EOMs, while a bit surprising, should have been anticipated because there isalways going to be a sector that produces the metric (2.4) − and the corresponding G-fluxcomponents to support it − as a solution to some EOMs. The reason is simple: the back-ground (2.4) appears from the most probable value in the generalized Glauber-Sudarshanstate. Such a system should be supported by minimizing an action if it has to survive ineleven (or ten in IIB)-dimensional space-time. The only action that we have here is (3.1),so it is not much of a surprise that we get (3.83). However what is surprising that theSchwinger-Dyson’s equations lead to two other set of equations: (cid:88) σ (cid:48) (cid:54) = σ (cid:28) δ S tot δ { g MN } (cid:29) ( σ | σ (cid:48) ) = (cid:28) δ S ghost δ { g MN } (cid:29) ( σ ) − (cid:28) δδ { g MN } log (cid:16) D † ( σ ) D ( σ ) (cid:17)(cid:29) σ (3.84) (cid:88) σ (cid:48) (cid:54) = σ (cid:28) δ S tot δ { C MNP } (cid:29) ( σ | σ (cid:48) ) = (cid:28) δ S ghost δ { C MNP } (cid:29) σ − (cid:28) δδ { C MNP } log (cid:16) D † ( σ ) D ( σ ) (cid:17)(cid:29) σ , that cleanly isolates the ghost action from (3.83) so that it appears only in (3.84). TheLHS of (3.84) requires us to take expectation values over states that generically do notlead to de Sitter spaces (as we saw in some simple computations above). More so, theLHS could even be complex. The RHS has ghost action that are expressed using fermionicvariable, whose variations with respect to the metric and the flux components could leadto complex quantities. The remaining terms should be balanced by the variations of the– 82 –og (cid:16) D † ( σ ) D ( σ ) (cid:17) part (note that D ( σ ) is not unitary). Despite encouraging signs of con-sistency, the two set of equations in (3.84) are in fact very hard to verify because of ourignorance of the complete behavior of either the ghost action or the displacement operator D ( σ ) from (2.61). Luckily however, the consistency of our analysis do not rely much onthe solutions of (3.84), as long as (3.83) has solutions. We will therefore leave the analysisof (3.84) for future work and concentrate on the solutions of (3.83).Finding the solutions to (3.83) is made easier because of our earlier works [21] and[22], where we studied the EOMs in great details. Since the readers could get most ofthe analysis from these two papers, we don’t want to repeat them here. Instead we wouldlike to emphasize the role played by the non-perturbative terms that shape the solutionsof (3.83) using the computations of section 3.2. To see this we will start with one of thespace-time EOM: δ S ( σ )tot δ (cid:104) g (cid:105) σ = (cid:104) R (cid:105) ( σ | σ ) − (cid:104) g (cid:105) σ (cid:104) R (cid:105) ( σ | σ ) − (cid:104) T (f)00 (cid:105) ( σ | σ ) − (cid:104) T (b)00 (cid:105) ( σ | σ ) − (cid:104) T (p)00 (cid:105) ( σ | σ ) − (cid:104) T (np)00 (cid:105) ( σ | σ ) = 0 , (3.85) where ( σ | σ ) denote the expectation values are taken over the generalized Glauber-Sudarshanstates σ ≡ ( α, β ) with the condition that the intermediate states are also σ . (cid:104) T (q)00 (cid:105) ( σ | σ ) is the expectation value of the energy-momentum tensors for q = (f , b , p , np), i.e. fluxes,branes, perturbative and non-perturbative quantum corrections respectively. The first twoterms of (3.85) then becomes: (cid:104) R (cid:105) ( σ | σ ) − (cid:104) g (cid:105) σ (cid:104) R (cid:105) ( σ | σ ) = R − g R = 1 g s d ( y ) + (cid:88) n ≥ f n ( y ) g n/ s η , (3.86) where the middle bold-faced terms are the Ricci curvature, metric and the Ricci scalarcomputed for the background (2.4) and are therefore g s dependent . On the extreme right,we show the g s dependence of these terms with ( d ( y ) , f n ( y )) being some functions of thecoordinates of the base manifold M × M of (2.3) and may be extracted from section4.1.4 (case 1, equations (4.69) and (4.70)) of [21]. For us, we will only worry about ( d , f ),and they take the following values: d ≡
3Λ + R2 H , f ≡ − H (cid:20) ( ∂H ) − (cid:3) H H (cid:21) , (3.87)where H ( y ) = h / ( y ) is the warp-factor in (2.4); Λ is the cosmological constant; ( ∂H ) = ∂ M H∂ M H with M ∈ M × M ; (cid:3) is the Laplacian over the internal manifold; and Ris the Ricci scalar computed using the internal metric components ( g mn ( y ) , g αβ ( y ) , g ab ( y )) without any g s or H ( y ) dependences. Both the terms in (3.87) have inverse g s dependences.Interestingly, as already pointed out in [21], the inverse g s dependence is very important. It This is a bit subtle. From (2.79) we know that (cid:104) g (cid:105) σ is not just g from (2.4), but has O (cid:16) g as M bp (cid:17) corrections (see footnote 26 and 35). However since the space-time metric, that goes as g − / s , is dominantover any perturbative corrections, we can safely ignore it here. We will come back to this point soon. – 83 –s easy to show that both the energy-momentum tensors for the G-flux components and forthe M2-branes, the inverse g s terms appear naturally (see for example equations (4.71) and(4.74) of [21]), so the question is to find them for the perturbative and the non-perturbativequantum terms. This is where the hard work of sections 3.1 and 3.2 pays off.The perturbative corrections to the energy-momentum is easier to handle so we willaddress it first. In some sense we already have the answer in (3.12). Here we want to seethe lowest order, i.e. n = 0 case in (3.86). This would be the choice q = 2 in (3.12), whichmeans: (cid:104) T p00 (cid:105) ( σ | σ ) ≡ C (2 , p )00 , (3.88)where p determines the moding of the G-flux components as given in (2.93) (see [21] formore details on this). It turns out, p ≥ , so the choice of q = 2 in (3.12) only provides acountable number of terms because of the following constraint: N + N + ( p + 2) N + (2 p + 1) N + ( p − N = 2 , (3.89)where N i ∈ Z are defined as in (3.8). In our case it appears that the contributions from(3.88) can only renormalize the contributions from the G-fluxes. This was already noticedin [21], so it is not a new observation. It is also easy to see that for p = 0, i.e. for time-independent G-flux components, there are an infinite number of solutions to (3.89) withno g s hierarchy. Unfortunately such terms also lack M p hierarchy as was shown in [22]with the choice of Ω ab as in (3.48), implying that there may not be an effective field theorydescription with time- independent G-flux components for the background (2.4).The non-perturbative corrections are more interesting, for they involve non-trivialcontributions from the quantum series (3.2). As we studied in sections 3.2.1 and 3.2.3, thenon-perturbative corrections typically appear from the BBS and KKLT type instantons,that have their roots in the non-local counter-terms that we discussed in [21]. There arealso non-perturbative corrections from fermionic condensates discussed in section 3.2.2. Allof these may be combined together to form: (cid:104) T (np)00 (cid:105) ( σ | σ ) ≡ T (BBS)00 + T (dBBS)00 + T (KKLT)00 + T (dKKLT)00 + T (ferm)00 , (3.90)where dBBS and dKKLT stand for delocalized BBS-type and delocalized KKLT-type in-stanton contributions to the energy-momentum tensors, and the last term in (3.90) is thefermionic contribution. All of their contributions, except the fermionic one, have been sum-marized in Table 4 , so we can easily extract what we need for our case. In the followingwe will denote the g s scalings of each of these contributions: T (BBS)00 = T (np;6)00 ( z, θ ) δ (cid:18) θ − − n (cid:19) , T (dBBS)00 = T (np;5)00 ( z, θ ) δ (cid:18) θ − − n (cid:19) T (KKLT)00 = T (np;7)00 ( z, θ ) δ (cid:18) θ − − n (cid:19) , T (dKKLT)00 = T (np;3)00 ( z, θ ) δ (cid:18) θ − − n (cid:19) , (3.91) where the explicit expressions for T (np;6)00 ( z, θ ) , T (np;5)00 ( z, θ ) , T (np;7)00 ( z, θ ) and T (np;3)00 ( z, θ )appear in (3.55), (3.54), (3.56), and (3.50) respectively; and θ is one-third the expression– 84 –n (3.89). All of these contribute to order g − s to the non-perturbative energy-momentumtensor (3.90), with n being the number of derivatives along direction M in (2.3). Thenumber of quantum terms contributing may again be extracted from the higher powers ofthe G-flux and the curvature tensors in (3.2), with the constraint: N + N + ( p + 2) N + (2 p + 1) N + ( p − N = 2 n + 2 l, (3.92)where l = 4 for the BBS-type instantons, l = 2 for the delocalized BBS-type and thedelocalized KKLT-type instantons, and l = 1 for the KKLT-type instantons. All of thesedepend on n , the number of derivatives along M , and this could be arbitrary. Howeverbecause of the exponential suppressions for higher values of n , as discussed in footnote 54,the series in n is convergent. The series in k , as they appear in the individual expressions ofthe non-perturbative energy-momentum tensors, is also convergent as discussed in footnote50. Thus in the end, the non-perturbative instantons contribute finite quantum correctionsto the Schwinger-Dyson equation (3.85).The fermionic contributions mostly appear from (3.43). However it will be interestingto bring it in the form of an expectation value over the coherent states σ = ( α, β, .. ) wherethe dotted terms now include the generalized Glauber-Sudarshan states for fermions. Thesefermionic coherent states are easy to construct in the same vein as the bosonic states, butwe will not do so here. It will suffice to know that they exist and contribute in the sameway as before. In fact existence of such a generalized Glauber-Sudarshan states will imply: (cid:104) ¯ΨΨ (cid:105) σ = (cid:88) p,p (cid:48) ¯ Ψ ( p ) Ψ ( p (cid:48) ) (cid:16) g s H (cid:17) ( p + p (cid:48) − / , (3.93)which may be derived from (3.38), and ¯Ψ ( p ) has no g s dependence. The readers should alsonote the notational diffence: ¯ΨΨ on the LHS of (3.93) is the fermionic condensate over thesolitonic background (2.1), whereas ¯ ΨΨ and ¯ Ψ Ω MN ab Ψ are the fermionic bilinears overthe background (2.4). In fact (3.93) is all we need to interpret the fermionic contributionsto the energy-momentum tensor in (3.43). All the fermionic bilinears appear from thecorresponding condensates over the generalized Glauber-Sudarshan states and contributeas: T (ferm)00 = T (np;2 b )00 ( z, θ ) δ (cid:18) θ k − (cid:19) , (3.94)where θ k as in (3.42) and the functional form for T (np;2 b )00 ( z, θ ) is given in (3.43). As notedearlier, (3.43) or (3.94) can now allow contributions like (cid:0) ¯ ΨΨ (cid:1) q , for q ≥ n + n + N + ( p − N + 2 q ( p −
2) = 4 , (3.95)with N i as in (3.8), and ( n , n ) are the derivatives along M and M respectively. The difference alluded to above is not just the difference in form of (3.95), but also the choice– 85 –or p which now takes p ≥ compared to p ≥ earlier. Interestingly, as noted in section3.2.2, when q = 0, p is bounded below by p ≥ .On the other hand, contributions directly from F-theory seven-branes appear from(3.32) only for the right embeddings, otherwise they only contribute perturbatively. Non-trivial embeddings will require the seven-branes to have some orientations along the ( a, b )directions. If this is the case, one has to make sure that such embeddings are stable , and donot revert back to the standard embeddings. For our case we will avoid complicating theanalysis, and only take standard embeddings of the seven-branes. As such (3.32) do notcontribute non-perturbatively. This would imply what we have so far should be enough toanalyze the Schwinger-Dyson equation (3.85), except that we will need one more definitionbefore we write down the space-time EOM. This is the G-flux expectation value: (cid:104) G MNPQ (cid:105) σ = (cid:82) [ D G MNPQ ] e i S tot D † ( σ ) G MNPQ ( x, y, z ) D ( σ ) (cid:82) [ D G MNPQ ] e i S tot D † ( σ ) D ( σ ) (3.96)= G MNPQ ≡ (cid:88) p,q G ( p,q )MNPQ ( y ) (cid:16) g s H (cid:17) p/ exp (cid:32) − qH / g / s (cid:33) + O (cid:18) g cs M bp (cid:19) , where p ≥ and q ≥ , N) ∈ M ×M × T G in (2.3). The result is straight-forwardbut some care is needed to interpret all the sides of (3.96). The LHS is the expectationvalue of the operator G MNPQ over the generalized Glauber-Sudarshan states σ ≡ ( α, β ).The path integral formula on the RHS has only fields, so G MNPQ is a field integrated in thestandard way with one key difference: the modes are to be selected from Υ k now, or moreappropriately the γ k ( x, y, z ) modes from (2.93), instead of the ones from (2.7). The part of D ( σ ) that is relevant for the computation in (3.96) is D ( β ), and S tot is the total action from(3.63) but one has to use the variables for the solitonic background (2.1). The second line isvery close to the expected G-flux components that we want for supporting a backgroundlike (2.4), with the additional g s corrections that are sub-dominant for g s <
1. Note thatthe expectation value leads to time-dependent G-flux components, reinforcing our earlierconclusion that the time-dependences of the expectation values and the existence of theGlauber-Sudarshan states go hand in hand.The energy-momentum tensors from the fluxes then follow similar path laid out earlieronce (3.96) is established. For us the concern is (cid:104) T (f)00 (cid:105) ( σ | σ ) . Using the input (3.96), theexpectation value of the energy-momentum tensor takes the following form: (cid:104) T (f)00 (cid:105) ( σ | σ ) = (cid:88) n (cid:32) h (1) n ( y ) + h (2) n ( y ) g s + h (3) n ( y ) g s (cid:33) η g n/ s , (3.97)where n ≥ h ( q ) n may be derived from [21]. Since we arelooking for g − s scalings, we only require the functional forms for h (2)0 ( y ) and h (3)3 ( y ), ormore appropriately, the functional form for the sum h (2)0 ( y ) + h (3)3 ( y ). This is easy to workout, and the result is: Following the notations of [21], and the fact that for q > g s →
0, we will only consider the components G ( p, ≡ G ( p )MNPQ in all our analysis, unless mentionedotherwise. – 86 – (2)0 ( y ) + h (3)3 ( y ) = − H (cid:16) G (3 / mnab G (3 / mnab + 2 G (3 / mαab G (3 / mαab + G (3 / αβab G (3 / αβab (cid:17) , (3.98) where ( m, n ) ∈ M ; ( α, β ) ∈ M and ( a, b ) ∈ T G . The other G-flux component donot participate to this order in g s . This is all consistent with what we had in [21], andmore so, we seem to be getting everything from expectation value over the generalizedGlauber-Sudarshan states. Finally, we can also add up all the perturbative and non-perturbative quantum terms from (3.88), (3.91) and (3.94) to get the final expression fortheir contributions to the energy-momentum tensors: (cid:104) T (Q)00 (cid:105) σ ≡ (cid:104) T (p)00 (cid:105) ( σ | σ ) + (cid:104) T (np)00 (cid:105) ( σ | σ ) = T (np;6)00 δ (cid:18) θ − − n (cid:19) + T (np;7)00 δ (cid:18) θ − − n (cid:19) (3.99)+ (cid:16) T (np;3)00 + T (np;5)00 (cid:17) δ (cid:18) θ − − n (cid:19) + T (np;2 b )00 δ (cid:18) θ k − (cid:19) + C (2 , p )00 , where θ is one-third the expression in (3.89); and n is the number of derivatives along M of (2.3). The most prominent contribution comes from the first term in (3.99), whichis from the BBS type instanton gas; and as we go to larger values of n , the contributionsbecome increasingly smaller. At each level of n , there are finite (and hence countable)number of terms, so the system is very well defined. Therefore plugging (3.99), (3.98),(3.87) and (3.86), in the Schwinger-Dyson equation (3.85), we get: δ S ( σ )tot δ (cid:104) g (cid:105) σ = 6Λ + R H − (cid:3) H H + η (cid:104) T (Q)00 (cid:105) σ + 2 (cid:16) h (2)0 + h (3)3 (cid:17) − ( n + ¯ n ) H √ g δ ( y − y o ) = 0 , (3.100) where R and g are respectively the Ricci scalar and the metric-determinant of the internalsix-manifold M × M from (2.3) without the warp-factor H or the g s factors, and Λ is thefour-dimensional cosmological constant in the IIB side. The last term appears from M2and M2-branes with the coefficients defined as in (3.1). The equation (3.100) is an exact equation in the sense that almost all possible contributions have been taken into account.In the following we will try to quantify briefly the above statement as most of this wasalready demonstrated rigorously in [21] (see for example the discussions in section 4.3.1 of[21]). However one puzzle appears now that has to do with the form of the expectationvalues in say (2.79), (3.96) and in the following: (cid:104) g αβ (cid:105) σ = F ( t ) g αβ ( y ) H ( y ) g − / s + .. = (cid:88) k ≥ D k (cid:16) g s H (cid:17) k − / H / ( y ) g αβ ( y ) + O (cid:18) g cs M bp (cid:19) (cid:104) g mn (cid:105) σ = F ( t ) g mn ( y ) H ( y ) g − / s + .. = (cid:88) k ≥ C k (cid:16) g s H (cid:17) k − / H / ( y ) g mn ( y ) + O (cid:18) g cs M bp (cid:19) , (3.101) where the corrections are sub-leading in the limit g s <
1. The analysis follows exactlythe same procedure we applied for (cid:104) g µν (cid:105) σ in (2.77) and (2.79) (the difference σ from α isirrelevant as we are in the gravitational sector). The aforementioned puzzle here is thatthe expectation values themselves have O (cid:16) g as M bp (cid:17) corrections taking us away, albeit in thesub-leading sense, from an exact de Sitter background. How are these corrections accom-modated in the Schwinger-Dyson equations? Additionally, how is the four-dimensionalNewton’s constant in the IIB side kept time-independent?– 87 –his is subtle so we need to tread carefully. The O (cid:16) g as M bp (cid:17) corrections that appear to theexpectation values, for example to (2.79), (3.96) and (3.101), come from the wave-functionof the interacting vacuum in (2.65). For the present purpose, we can generalize it toΨ ( σ )Ω ( g MN , C MNP , t ). The wave-function has three parts: (a) the Glauber-Sudarshan wave-function for the harmonic-vacuum, (b) the correction coming from δ D and (c) the integrated effect from the full interacting Hamiltonian H int . The integration is from − T = −∞ (ina slightly imaginary direction) till the present epoch t (or √ Λ t ). On the other hand, theSchwinger-Dyson equation, in say (3.100), makes sense only for − √ Λ < t <
0. Whathappens in these two regimes?One answer could be that the integrated effect on the wave-function (2.65) cancels outcompletely so that there are no O (cid:16) g as M bp (cid:17) corrections to either (2.4) or to (3.96). Such aconclusion would be consistent with the result we get from the Schwinger-Dyson equation,and in turn will confirm the outcome (1) given towards the end of section 2.4.Unfortunately such a conclusion is very hard to prove because we have no control onthe dynamics for t < − √ Λ . All we can say here is that the integrated effect of H int onthe wave-function (2.65) appears to cancel out in the regime − √ Λ < t <
0. This will leadto the outcome (2) in section 2.4, although there is a possibility is that maybe (3.83) isnot completely correct. Could it be possible that the Schwinger-Dyson equations (3.68)actually decompose to the following: δ S ( σ )tot δ (cid:104) g MN (cid:105) σ = (cid:28) δ S ghost δ { g MN } (cid:29) ( σ ) − (cid:28) δδ { g MN } log (cid:16) D † ( σ ) D ( σ ) (cid:17)(cid:29) σ − (cid:88) σ (cid:48) (cid:54) = σ (cid:28) δ S tot δ { g MN } (cid:29) ( σ | σ (cid:48) ) (3.102) δ S ( σ )tot δ (cid:104) C MNP (cid:105) σ = (cid:28) δ S ghost δ { C MNP } (cid:29) σ − (cid:28) δδ { C MNP } log (cid:16) D † ( σ ) D ( σ ) (cid:17)(cid:29) σ − (cid:88) σ (cid:48) (cid:54) = σ (cid:28) δ S tot δ { C MNP } (cid:29) ( σ | σ (cid:48) ) , instead of (3.83)? There is something interesting about the set of equation in (3.102): theextra O (cid:16) g as M bp (cid:17) corrections to the expectation values (2.79), (3.96) and (3.101) can now becompensated by the RHS of (3.102) such that (3.100) remains unchanged without any extra O (cid:16) g as M bp (cid:17) factors. All EOMs that we studied in [21] remain as they were without any extrafactors. This is good, but there is an issue with (3.102) when we compare with (3.70). Theterms of (3.70) starts with warped metric and G-flux components. If we look at the g part of the solution, then the LHS of (3.102) starts with terms with g s dependence as g − s ,whereas the corresponding terms in (3.70) start with g − / s . Thus they cannot be matched.While this doesn’t disprove the existence of an equation like (3.102), the non-existence of (3.83) would be much more puzzling: even if the metric and the G-flux componentsreceive O (cid:16) g as M bp (cid:17) corrections, the corrected metric and the flux components should satisfyequations similar to (3.83), otherwise there is a possibility that such configurations cannotbe supported in eleven-dimensional space-time. All of these then appears to indicate that(3.102) cannot be the right EOMs, and our earlier EOMs, (3.83) and (3.84), should stillbe the correct ones here. This is despite the fact that we never realize a configuration like(2.4), and the corresponding G-flux components, by directly solving supergravity EOMshere. – 88 –e should then look for a solution to the conundrum that not only allows SDEslike (3.83), but also shows that there may not be any extra O (cid:16) g as M bp (cid:17) corrections to theexpectation values themselves. To find this, let us go back to the functional form for a eff and a † eff from (2.59). The effective annihilation operator a eff was defined via the action a eff | Ω (cid:105) = 0, i.e. a eff annihilates the interacting vacuum | Ω (cid:105) . However as discussed infootnote 23, this is not enough to fix the form for a eff unambiguously implying, in turn,that the form for the displacement operator D ( σ ), where we use σ = ( α, β, .. ) instead of α ,described via (2.57) or (2.61), cannot be fixed unambiguously either. More importantly,it is δ D ( σ ), defined in (2.64) that suffers ambiguity because D , which is the displacementoperator for the harmonic vacuum, or ˆ D ( σ ), which is the non-unitary version of D ( σ ),are unambiguously fixed. The functional form for a eff is defined by c lmn parameters, andif each ( l, m, n ) go from 1 to N , where N is arbitrarily large, then there are at least N number of parameters here. We can make this more precise by assigning the followingfunctional form for a eff : a eff ( k ) = a k + (cid:88) { i p } (cid:90) c i ....i n ( k ; k , .., k n ; t ) n (cid:89) i =1 a ( i i ) k i δ (cid:32) n (cid:88) l =1 k l − k (cid:33) d k i + O [exp ( c i ....i n )] (3.103)+ (cid:88) { j p } (cid:90) d j ....j m ( k ; k , .., k m ; t ) m (cid:89) j =1 a † ( j i ) k i δ (cid:32) n (cid:88) l =1 k l − k (cid:33) d k j + O [exp ( d j ....j n )] + permutations , where the permutations involve various symbolic permutations of the a ( j i ) k i of a † ( j l ) k l both inthe polynomial and the exponential forms, and the j i superscript denote the creation or theannihilation operator for a given component of g MN or C MNP . The coefficients c i ....i n and d i ....i n are functions of k i and t and also of the string coupling in the solitonic background,and thus should be related to c lmn in (2.59). The above form (3.103) is the most genericannihilation operator one could write for an interacting theory, although one can easilysee that imposing a eff ( k ) | Ω (cid:105) = 0 cannot fix the forms of c i ....i n and d i ....i n unambiguously.Interestingly however, the number of variables appearing in (3.103) seems to be similarto the number of variables that would appear in the interacting Hamiltonian H int . Thismeans, δ D ( σ ) defined as (2.64) will also have exactly the same number of variables as in H int , and we can, in turn, use this information to fix the form of δ D ( σ ) as: (cid:90) [ D g (cid:48) MN ][ D C (cid:48) PQR ] (cid:104) g MN , C PQR (cid:12)(cid:12) δ D ( σ ( t )) (cid:12)(cid:12) g (cid:48) MN , C (cid:48) PQR (cid:105) Ψ ( g (cid:48) , C (cid:48) ) = − (cid:88) n ( − i ) n n ! (cid:90) (cid:2) D g (cid:48) D ˆ g (cid:3) MN (cid:104) D C (cid:48) D ˆC (cid:105) PQR × (cid:104) D ( σ ( t )) (cid:105) (cid:90) t − T dt .....dt n (cid:104) ˆ g MN , ˆC PQR (cid:12)(cid:12) T (cid:40) n (cid:89) i =1 (cid:90) d x i H int ( t i , x i , y i , z i ) (cid:41) (cid:12)(cid:12) g (cid:48) MN , C (cid:48) PQR (cid:105) Ψ (cid:0) g (cid:48) , C (cid:48) (cid:1) , (3.104) with T → ∞ in a slightly imaginary direction and we have defined (cid:104) D ( σ ( t )) (cid:105) as (cid:104) D ( σ ( t )) (cid:105) = (cid:104) g MN , C PQR | D ( σ ( t )) | ˆ g MN , ˆC PQR (cid:105) and Ψ ( g (cid:48) , C (cid:48) ) ≡ Ψ (g (cid:48) MN , C (cid:48) PQR ) is the vacuum statewave-function. In the second line we have used D ( σ ) instead of D ( σ ) because both δ D ( σ )and H int are already proportional to powers of the string coupling, so δ D ( σ ) H int wouldbe highly sub-leading. The equation (3.104) can be exactly solved despite the complicated– 89 –ature of it, and we get : D ( σ, t ) = D ( σ, t ) exp (cid:18) i (cid:90) t − T d x H int (cid:19) (3.105)where H int is the full interacting Hamiltonian that we discussed in sections 3.1 and 3.2,and T → ∞ (1 − i(cid:15) ). The above identification easily justifies δ D ( σ ) to be proportionalto powers of H int with no zeroth order terms. The degrees of freedom also match, and(3.105) satisfies (3.104) to all orders in string coupling. The wave-function of the shiftedinteracting vacuum now satisfies:Ψ ( α )Ω ( g µν , t ) = exp (cid:20)(cid:90) + ∞−∞ d k log (cid:16) Ψ ( α ) k ( (cid:101) g µν ( k ) , t ) (cid:17)(cid:21) , (3.106)which is exactly the Glauber-Sudarshan wave-function. For computational purpose, es-pecially in the path integrals, we can replace D ( σ ) in (3.105) by the non-unitary partˆ D ( σ, t ), as we have done so earlier. In fact including (3.105) in the path integral com-putation, say in (2.71), (2.78) and (2.82), one can easily show that there are no O (cid:16) g as M bp (cid:17) corrections to the results anymore. This would extend to (3.101) too, and therefore (2.4)will continue to be the exact answer that we get from the Glauber-Sudarshan states in theinterval − √ Λ < t <
0, leading to option (2) given towards the end of section 2.4.The exactness alluded to above suggests that the lowest order EOMs appearing fromthe SDEs should fix the form of both the de Sitter metric as well as the internal manifold.How is this possible in the light of higher order g s corrections? The answer is not too hardto see. Consider for example (3.86). The RHS of the equation is determined from d ( y )and f n ( y ), and the lowest order EOMs fix the forms of d ( y ) and f ( y ) in (3.87). Imaginethis fixes the Ricci scalar R and the cosmological constant Λ (of course other lowest orderEOMs should participate to achieve the goal). Once we go to the next order in g s , thefunction f n ( y ) in (3.86) develops higher order g s corrections from the F i ( t ) factors in (2.2)or the M-theory uplift (2.4). We therefore conclude the following:The fact that the full non-K¨ahler internal metric over the space M ×M appears from tak-ing the expectation values over the generalized Glauber-Sudarshan states is a consequenceof two underlying conspiracies: one, the choice of the modes (cid:16) η k ( x , y, z, t ) , ξ k ( x , y, z, t ) (cid:17) along directions M and M respectively; and two, the choice of the Glauber-Sudarshanstates | σ (cid:105) ≡ D ( σ ) | Ω (cid:105) with | Ω (cid:105) being the full interacting vacuum in M-theory, and D ( σ )satisfying (3.105).Putting (2.79) and (3.101) together, leads to the emergence of the full metric (2.4) fromexpectation values over these states. Finally, the coefficients C k and D k in (3.101), maybe easily derived from the following equation: (cid:88) { k i } D k C k C k (cid:16) g s H (cid:17) k + k + k ) / = 1 , (3.107) There is of course a constant of proportionality accompanying (3.105) which is the overlap integral (cid:104) Ω | (cid:105) that we ignore here. There has to be a non-zero overlap, but other than that this is essentially aconstant. – 90 –y going order by order in powers of g s /H with C = D = 1 and k ∈ Z . For examplegoing to next order g / s , we get D / = − C / , D = 3 C / − C , etc. All of thesekeep the four-dimensional Newton’s constant time-independent in the IIB side, once weimpose (3.105), although one question arises: What about renormalization or running ofthe four-dimensional Newton’s constant? Could this happen here?To answer this and other related questions, we have to get back to the discussion thatwe left-off before (3.101) and ask what happens once we go to higher orders in g s . First, f n ( y ) contains all informations of the F i ( t ) factors, so higher order in g s will switch onhigher order terms in C k and D k from (3.101). In fact we are looking at terms that scaleas (cid:0) g s H (cid:1) ( n − / with n >
0. The quantum contributions to the energy-momentum tensor toany n can be written from (3.99) as: (cid:104) T (Q | n )00 (cid:105) σ ≡ (cid:104) T (p | n )00 (cid:105) ( σ | σ ) + (cid:104) T (np | n )00 (cid:105) ( σ | σ ) = T (np;6)00 δ (cid:18) θ − n − n (cid:19) + T (np;7)00 δ (cid:18) θ − n − n (cid:19) + (cid:16) T (np;3)00 + T (np;5)00 (cid:17) δ (cid:18) θ − n − n (cid:19) + T (np;2 b )00 δ (cid:18) θ k − n (cid:19) + C ( n +2 , p )00 , (3.108) where we see that as we go to higher n , the quantum terms become increasingly moreinvolved, although now there are two suppression factors: higher n are suppressed bypowers of g s , and higher n are suppressed by exponentially decaying factors. The storyshould now be clear. As we go to higher order in g s , (a) higher orders in G-fluxes, i.e. p > in (3.96), (b) higher orders in F i ( t ) factors, i.e. ( C k , D k ) for k > i.e. n > simultaneously switchedon. The equations, up to the next two orders in g s , governing these modes are now:2 η T ( Q | = η ii T ( Q | ii , C / = 3 (cid:16) η T ( Q | − η ii T ( Q | ii (cid:17) , (3.109)where the repeated indices are summed over. The above two trace equations imply thatthe higher order quantum terms are balanced against the higher order terms from F i ( t )factors, keeping the lowest order background (2.4) intact . The above two equations alsoimply delicate balancing as C / is a constant but the quantum terms are classified by(3.108). The fluxes, to this order, cancel out, so they do not contribute to the traceequations. Similar story unfolds along the ( m, n ) directions because the quantum termstherein are of the form: (cid:104) T (Q | s ) mn (cid:105) σ = T (np;6) mn δ (cid:18) θ − s − n (cid:19) + T (np;7) mn δ (cid:18) θ − s − n (cid:19) (3.110)+ (cid:16) T (np;3) mn + T (np;5) mn (cid:17) δ (cid:18) θ − s − n (cid:19) + T (np;2 b ) mn δ (cid:18) θ k − s (cid:19) + C ( s +2 , p ) mn , with s ≥
0; and where T (np; r ) mn for r = 6 , , , b are defined in (3.55), (3.54), (3.56),(3.50), and (3.43) respectively. The other two variables ( θ k , θ ) are in (3.42) and one-third the function in (3.89) respectively. Note, despite similar classification with respect to( θ, θ k ), the quantum terms are in general different from (3.108). Similarly the contributionsfrom the G-fluxes are also different from (3.97) and (3.98); and may be written as: (cid:104) T (f) mn (cid:105) ( σ | σ ) = (cid:88) s ≥ (cid:32) T (1 | s ) mn ( y ) + g s T (2 | s ) mn ( y ) + T (3 | s ) mn ( y ) g s (cid:33) g s/ s , (3.111)– 91 –here the functional form for T ( r | s ) mn ( y ) may be extracted from eq. (4.12) of [21]. Interest-ingly now, because of the fact that p ≥ in (3.96), the lowest order s = 0 contributionsonly appear from T (3 | mn ( y ) and not from T (1 | mn ( y ). Therefore combining (3.110) and (3.111)together, we get the SDE δ S ( σ )tot δ (cid:104) g mn (cid:105) σ = 0 satisfied by the unwarped ( i.e. g s and H ( y ) inde-pendent) internal metric component g mn along M in (2.3) as:R mn − g mn R − H g mn = (cid:104) T ( Q | mn (cid:105) σ + T (3 | mn , (3.112)where the last two terms appear from (3.110) and (3.111) respectively, H ( y ) is the warp-factor; and Λ is the cosmological constant. The metric g mn ( y ) is clearly non-K¨ahler withthe Ricci scalar satisfying the relation R = − (cid:0) T ( Q | + T (3 | (cid:1) − H where the termin the bracket is the trace of the RHS of (3.112). This means some part of the internalcurvature does get contribution from the four-dimensional cosmological constant Λ in theIIB side.What happens when we go to higher orders in g s ? Here it would mean going to orders g / s , g / s and beyond. Something interesting happens now. To the two higher orders in g s , the Schwinger-Dyson equations reveal the following two equations: g mn = 1 A ( y ) (cid:18) (cid:104) T ( Q | mn (cid:105) σ + T (3 | ) mn (cid:19) , g mn = 1 A ( y ) (cid:16) (cid:104) T ( Q | mn (cid:105) σ + T (3 | mn (cid:17) , (3.113)where A ( y ) and A ( y ) are two functions that may be read from eq (4.19) and eq (4.24)respectively of [21]. On the RHS of the two equations, both the quantum and the fluxterms are at higher orders. The quantum terms are (cid:104) T ( Q | mn (cid:105) σ and (cid:104) T ( Q | mn (cid:105) σ from (3.110);and the flux terms are T (3 | ) mn and T (3 | mn from (3.111). Whereas on the LHS are the zeroth order unwarped metric components. If we go to even higher orders in g s , we get similarequations. This implies that as we go to higher order in g s , higher order terms in quantum,G-flux and F i ( t ) are switched on in such a way that the unwarped metric g mn remainsintact. This is our stability criterion and it occurs in the following way.The higher order G-flux components with p > in (3.96) and higher order F i ( t ) componentswith k > balance against the higher order quantum terms, for example with n ≥ s ≥ all orders in g s and M p such that the background (2.4) along-with the supporting G-flux componentsremain uncorrected to arbitrary orders in g as M bp provided the choice (3.105) is considered.Such a balancing criterion is interesting but question is what it implies for the runningof the four-dimensional Newton’s constant? The four-dimensional Newton’s constant isof course time-independent for the solitonic vacuum, but for the background (2.2), or it’sM-theory uplift (2.4), it depends crucially on the un-warped metric components g mn and g αβ (recall F ( t ) F ( t ) = 1 so it introduces no time dependence). Our discussion aboveshows that both the internal components of the metric do not receive O (cid:16) g as M bp (cid:17) corrections.Does that mean the four-dimensional Newton’s constant do not get renormalized? The– 92 –nswer turns out to be the opposite: there does appear to be finite renormalization of theNewton’s constant. To see this let us go back to the metric equation (3.112). On the RHSthere are flux contributions from T (3 | mn and quantum contributions from (cid:104) T ( Q | mn (cid:105) σ . Thequantum contributions to the energy-momentum tensor are determined by ( θ, θ k ) that canbe easily read up from (3.110). Once we know the values for θ k and θ , they will fix the number of quantum terms from (3.2) that contributes. What this does no tell us is the exact coefficients of the quantum terms contributing to the energy-momentum tensor. Since thesecoefficients appear from integrating out the high energy modes over the solitonic vacuum,their precise values might depend on what energy scale we are in. This means the RHS of(3.112) could have some dependence on the energy scale, implying that the metric factor,and therefore the volume of the internal six manifold (i.e the base in (2.3)), might have somescale dependence. From here it appears that the four-dimensional Newton’s constant couldin principle get renormalized accordingly (although it will be time-independent). Of coursethere is a possibility that the contributions of the quantum terms are such that the volumeof the six-manifold do not change, implying no renormalization of the four-dimensionalNewton’s constant. This would then be an interesting and surprising conclusion, althoughto verify either of these conclusions would require us to work out the precise coefficients ofthe quantum terms contributing to the energy-momentum tensor. Such a computation isclearly beyond the scope of this work, and will hopefully be dealt in near future. Let us briefly discuss how moduli stabilization could work in a set-up like ours. On the soli-tonic background, the metric configuration is given by (2.1). Let G (0)MNPQ ( y ) and G (0)0 ij M ( y ) − where (M , N) and ( i, j ) denote the coordinates of eight-manifold (2.3) and two spatial di-rections respectively − be the G-flux components to support the metric configuration (2.1).The system is governed by an interacting Hamiltonian H int which, as we saw earlier, hasan infinite number of local and non-local, including their perturbative, non-perturbativeand topological, interactions. Switching on such interactions would fix all the K¨ahler andthe complex structure moduli of the eight manifold (similar stabilization will occur on thedual IIB side also). Once the moduli are fixed at the solitonic vacuum, we can study thefluctuations and from there construct the Glauber-Sudarshan state. The metric and theG-flux components of the de Sitter space are then: (cid:104) g µν (cid:105) σ = g − / s η µν , (cid:104) g αβ (cid:105) σ = g − / s H ( y ) F ( t ) g αβ , (cid:104) g mn (cid:105) σ = g − / s H ( y ) F ( t ) g mn (cid:104) g ab (cid:105) σ = g / s δ ab , (cid:104) G MNPQ (cid:105) σ = (cid:88) p ≥ / G ( p )MNPQ (cid:16) g s H (cid:17) p , (cid:104) G ij M (cid:105) σ = (cid:88) p ∈ Z G ( p )0 ij M (cid:16) g s H (cid:17) ( p − , (3.114)where (M , N) are the coordinates of the eight-manifold, ( m, n ) ∈ M , ( α, β ) ∈ M and( a, b ) ∈ T G . It is also known that G (0)0 ij M = − ∂ M (cid:0) (cid:15) ij H (cid:1) . The above set of relations tell usthat the Glauber-Sudarshan state would allow the internal moduli to vary accordingly with g s and F i ( t ) in a controlled way described above, and there would be no Dine-Seiberg [31]runaway. This is what we referred to as the dynamical moduli stabilization earlier.– 93 –he next question is how to quantify the supersymmetry breaking in our set-up. Todo this we will have to work out the SDEs for the flux sector given in (3.83). We will notwork out all the flux equations here, as most are presented in [21], but will suffice ourselveswith one set of equations given by the following SDE: δ S ( σ )tot δ (cid:104) C (cid:105) σ = 0 . (3.115)The flux EOM from (3.115) is a bit more non-trivial to work out because we need to considerthe contributions of the quantum terms from the topological sector also. Nevertheless, aftersome careful manipulations, the result may be presented in the following way: − b (cid:3) H + 1 √ g (cid:88) { k i } ∂ N (cid:16) √ g H G ( k )012 M g MN (cid:17) C k D k δ ( k + k + k −
3) (3.116)= b (cid:88) { k i } G (3 / N ...N (cid:16) ∗ G (3 / (cid:17) N ...N + 1 √ g [ Y ] + b √ g (cid:88) { k } ∂ N (cid:16) √ g (cid:16) Y ( k )4 (cid:17) N (cid:17) δ (cid:18) θ − (cid:19) , where ( b , b ) are numerical constants and b ≡ (cid:80) { k i } C k D k δ ( k + k −
3) with ( C k , D k )are defined as in (3.101). Y is the eight-form defined in (3.83) and we take the simplifiedversion with a = a = a = ... = 0, with the index-free notation [ Y ] being defined asthe contraction of Y with the un-warped ( i.e. g s independent) epsilon tensor. Y ( k )4 arethe quantum terms from the topological sectors and are classified by θ = , where θ isone-third the function in (3.89). These topological terms may be formally presented in thesame way as we did in section 3.1, but we won’t do it here. The readers may look up ourearlier work [21] for details on this. Finally, the flux-component G ( k )012 M may be easily readup from (3.114).Let us now compare (3.116) with SDE from the gravitational sector, namely (3.100).Both these equations are written in terms of (cid:3) H and square of the G-flux components,which appears in (3.100) via (3.98). There are however few differences, which are crucial:the quadratic part of the G-flux components in (3.116) appear with a Hodge star, plus thequantum contributions are a bit different from (3.100). Multiplying (3.100) by b andsubtracting it from (3.116), will remove the (cid:3) H , and we can easily see that: (cid:12)(cid:12)(cid:12) G (3 / ab − ( ∗ G ) (3 / ab (cid:12)(cid:12)(cid:12) > , (3.117)signalling the breaking of supersymmetry; with (M , N) restricted to, and the Hodge stardefined using the un-warped metric of, M × M (i.e the metric components g mn and g αβ ).One could also express (3.117) as in (2.95) (or as in footnote 43), but both of these wouldeventually become (3.117). In the construction of Y ( k )4 we have only taken the infinite set of perturbative terms from (3.2). Thereis of course the whole non-perturbative sector, similar to what we had in section 3.2 and thus affecting thetopological interactions, that we do not consider in (3.116). Thus to compare the quantum terms of (3.116)with the ones in (3.100) we will have to introduce the non-perturbative corrections. This is technicallychallenging, but we do know that their contributions will be finite , just like what we had in (3.100). Moredetails on this will appear elsewhere. – 94 –e can quantify the supersymmetry breaking even further by analyzing the fermionicterms on the seven-branes as studied in section 3.2. Our aim here would be to show how(3.117) actually breaks supersymmetry. For this, let us consider the fermionic action from(3.41). The relevant terms may be arranged together from (3.41) to take the followingsuggestive form: S (cid:48) = T (cid:90) d σ √− g (cid:34) (cid:0) tr adj ¯ ΨΨ (cid:1) q G MN ab + tr adj ¯ Ψ (cid:16) e ˆΩ MN ab + e ˆΩ (cid:48) MN ab (cid:17) Ψ (cid:35)(cid:32) G MN ab − ( ∗ G ) MN ab (cid:33) , (3.118) where the bold faced fields are extracted from the expectation values as in (3.114), andtherefore ∗ is now defined with respect to the g s dependent metric components. Theother quantities appearing in (3.118) are defined from (3.37) and (3.35), and in fact thesecond term in the action appears from | G totMN ab | with G totMN ab as in (3.40). The choiceof the relative sign is motivated from [51] although the analysis here is very different (wealso have branes and not anti-branes here ). Taking q = 1 in (3.118) and dimensionallyreducing the above integrand to four space-time dimensions will provide a mass term tothe fermion coming from Ψ ( y m , g s ) with the mass term being proportional to (3.117) oncewe take the lowest order g s components from (3.114). On the other hand, over the solitonicbackground (2.1), the fluxes remain self-dual and we see that no mass term is generated.This mass term in the time-dependent case of course breaks supersymmetry, but here wesee that it also a function of the coordinates of the internal six-manifold M × M . Thusinstead of using the mass term for the fermions to contribute to the vacuum energy, we caninterpret (3.118) as another interaction in the theory. This way the contributions to thecosmological constant would only appear from the fluxes and the quantum terms, exactlyas we have advocated earlier (see for example eq. (4.192) in [21]).In determining the supersymmetry breaking condition in (3.117) and (3.118), we havekept a subtlety under the rug related to the connection to the cosmological constant Λ. Asdiscussed above, the cosmological constant is an emergent quantity in our model, meaningthat its value is determined by the fluxes, branes and the quantum corrections, and is nota quantity that we add to the EOMs by hand. On the other hand, the supersymmetrybreaking condition is also determined in terms of fluxes and quantum corrections as maybe seen by subtracting (3.100) from (3.116), or directly from (3.118). Does this meanthat the supersymmetry breaking scale is determined by the cosmological constant (or theHubble parameter)? The answer is no , because the cosmological constant appears froman integrated condition as shown in [3, 21] and is therefore suppressed by the unwarpedvolume of M × M in the following way: Λ = 112 V [ T ( Q ) ] ii − V H (cid:16) T ( Q ) ] aa + [ T ( Q ) ] MM (cid:17) − V H (cid:104)G (3 / ab G (3 / ab (cid:105) av − n T V H , (3.119) It is interesting to note here that the mass term coming from | G MN ab − ( ∗ G ) MN ab | could in principlebe related to switching on (0 ,
4) fluxes over the eight-manifold (2.3), much along the lines of [51, 57]. Theeight-manifold doesn’t have to be a complex manifold as long as it has an almost complex structure. Moredetails on this will be presented elsewhere. – 95 –here the repeated indices are summed over with (M , N) ∈ M × M and V is the un-warped volume of the six-manifold. We have taken the warp-factor H ( y ) = constant forsimplicity and ( n , T ) are the data for the integer and fractional M2-branes from (3.1).We have also defined: [ T ( Q ) ] MM ≡ (cid:90) d y √ g g MN (cid:104) T ( Q )MN (cid:105) σ , (cid:104)G (3 / ab G (3 / ab (cid:105) av ≡ (cid:90) d y √ g G (3 / ab G (3 / ab , (3.120) where g MN is the un-warped metric, and (cid:104) T ( Q )MN (cid:105) σ is the expectation value of the energy-momentum tensor over the Glauber-Sudarshan state | σ (cid:105) similar to what we showed in(3.99). This means they contain all the perturbative and the non-perturbative contribu-tions, and are thus classified accordingly. Additionally, the relative signs in (3.119) areimportant, and as long as the quantum terms along the two spatial directions dominate over all the other negative terms, the cosmological constant will be positive. The questionis how big it can be?In a moduli stabilized scenario, the un-warped volume of the six-manifold, V , is fixedto a large value so that supergravity description may be valid. As such this implies thatthe cosmological constant should be very small. However at this stage one might questionthe fact that both the G-flux components and the quantum terms also appear as integratedover the eight-manifold in (3.119). Shouldn’t the volume factors cancel out? The answeris again no , because the G-flux components that appear above are of the form G (3 / ab which are highly localized fluxes and therefore the global behavior do not effect them. Thequantum series are also constructed out of these flux and metric components (the lattercould also be taken to be localized functions) so there indeed appears a genuine volumesuppression in the expression of the four-dimensional cosmological constant Λ in (3.119).On the other hand, there is no such suppression factor in the supersymmetry breakingcondition, therefore it appears that the supersymmetry breaking scale should be muchlarger than the cosmological constant (or the Hubble scale). In addition to that, there areother differences, namely in the exact arrangement of the flux and the quantum terms in(3.117) and (3.119), confirming that the two quantities cannot be similar.Finally, let us ask what happens when we go to the strong coupling limit of typeIIB. Recall that our analysis is done in the type IIB side at the constant coupling limitof F-theory [56] where we allow constant dilaton and vanishing axion fields. We can nowS-dualize the IIB background which will simply change the type IIB metric by a constantfactor (if ϕ b denotes the constant dilaton in the IIB side, then under a S-duality the metricchanges by a constant multiplicative factor proportional to e − ϕ b , with the dilaton changingby ϕ b → − ϕ b ). This means we are dealing with exactly similar background as before!Lifting this to M-theory will then reproduce the metric and the flux components from theexpectation values over a similar Glauber-Sudarshan state just like we had earlier, implyingthat the type IIB strong coupling configuration mirrors the weak coupling scenario to agreat extent. All the conclusions in the presence of time-dependent degrees of freedom − and therefore the pathologies in the absence of time dependences − will carry over to thestrong coupling side as before. The quantum break time, i.e. where the type IIA strong– 96 –oupling sets in, will change to: − e ϕ b √ Λ < t < , (3.121)where t is now an appropriate scaled temporal coordinate that keeps the metric configu-ration unchanged (other spatial coordinates need to be scaled in a similar way). Thus atboth strong and weak coupling, time-dependent degrees of freedom appear to be essentialto allow for a four-dimensional EFT description to be valid, although it seems like the situ-ation at unit type IIB string coupling should remain out of the reach of our analysis. Thisis not quite so, because the type IIA coupling g s ≡ g b H Λ | t | << g b ≥ .All the above set of computations would hopefully convince the readers that theGlauber-Sudarshan state indeed captures all the essential properties of a de Sitter space. Inthe following section we will discuss other properties of the Glauber-Sudarshan state thatwill further reinforce the fact that representing de Sitter space as a Glauber-Sudarshan isnot only solid but appears to be essential to allow for a stable, non-supersymmetric stateto exist in string theory.
4. Properties of the Glauber-Sudarshan state
As mentioned in the Introduction, there has been a recent slew of papers arguing againstthe validity of long-lived de Sitter spacetimes from string theoretic derivations. Althoughthe so-called de-Sitter conjecture was motivated by the difficulty of finding meta-stable deSitter vacua in string theory [13], soon evidence for it came when starting from the distanceconjecture [60], and invoking the Bousso covariant entropy bound [58], for a causal patch inde Sitter spacetime [16]. The distance conjecture, having been tested more extensively instring theory constructions [61], put the de Sitter conjecture on a much firmer footing. Ithas since also been shown that one can arrive at the de Sitter conjecture starting from thedistance conjecture by assuming the species bound [62]. However, what was still lacking is adeeper quantum gravity argument, revealing the underlying reason why such a conjecture,claiming the absence of meta-stable de Sitter spacetimes in string theory, should be takenseriously. One such argument came in the form of the ‘no eternal inflation’ principle [63] In addition to that, imagine that the IIB coupling takes the maximum value at y = y , with y ∈M × M , and the warp-factor H( y ) peaks at y = y , then we can still be at the weak-coupling limitin the type IIA side as long as g s ≡ g b ( y )H ( y )Λ | t | <<
1. Thus, despite deviating away from theconstant-coupling scenario, the analysis may still be easily tractable. – 97 –nd another from the so-called ‘trans-Planckian censorship conjecture’ (TCC) [24]. Arelated argument also came in the form of the quantum breaktime of de Sitter spacetimes after which the interactions break down the semiclassical description of de Sitter spacewith a causal horizon [29]. Although these arguments differ amongst themselves regardingthe time of validity of a consistent meta-stable de Sitter spacetime (which we shall discusslater on), together they establish more fundamental evidence for the de Sitter conjecture,albeit at the cost of refining the original conjecture by allowing for short-lived meta-stablede Sitter spacetimes.In the following, we shall establish how our solution manages to escape the swamplandby focussing on the TCC since it is the most concrete principle from which the de Sitterconjecture can be derived . More generally, we shall also point out that the time scales onwhich our solution can be trusted is completely compatible with the allowed lifetime of a(short-lived) metastable de Sitter spacetime, as per the swampland. We shall then focus onhow radiative corrections, which typically forces one to fall into the quantum swampland [40], is also naturally avoided by our solution. Finally, we shall turn to old argumentswhere it was established that the symmetries of a de Sitter spacetime should necessarilybreak after some time, and not be eternal, for it to have a finite entropy and show how oursolution automatically complies with this restriction. Typically, the above-mentioned arguments lead to an upper bound on the lifetime of anyde Sitter vacua which is far shorter than those associated with stringy constructions. Oneof these arguments – the TCC [24] – is an elevation of the old ‘trans-Planckian problem’of inflationary cosmology [25] to the level of a hypothesis. To understand this problem indetail, and how our solution eventually manages to avoid it, let us recall that in free quan-tum field theory on Minkowski spacetime, one starts by canonically quantizing the fieldsdescribed on a Fock space. Even for an expanding background, cosmological perturbationscan be similarly quantized since at linear order ( i.e. only considering the quadratic Hamil-tonian), each of the Fourier modes evolve independently. Thus, ignoring non-Gaussianities,one needs to quantize a set of harmonic oscillators described on a Fock space, as in the caseof flat spacetime. However, the novelty lies in the fact that the mass of these oscillatorsare time-dependent due to the time-dependence of the dynamical background. In otherwords, the Fourier modes are quantized in terms plane wave modes which have a constantwavelength in comoving coordinates. However, this implies that the physical wavenumberof these modes are redshift with time, given by p = k/a ( t ), where a ( t ) is the scale factor ofthe universe and p and k stand for the physical and comoving wavenumber, respectively.The most commonly understood manifestation of the trans-Planckian problem occurswhen one takes a macroscopic classical perturbation today and ‘evolves’ it backwards intime. The physical wavelength associated with this mode gets blueshifted due to the ex-pansion of spacetime. If one allows for the quasi-de Sitter phase of expansion to last for a Indeed, the de Sitter conjecture is more vague and invokes some O (1) numbers which can be explicitlyfixed only when referring to the TCC [24] (or some similar principle). – 98 –ignificant long amount of time, one would find that a classical perturbation mode, visiblein the sky today, actually originated from a physical wavelength smaller than the (four-dimensional) Planck length (cid:96) Pl . Of course, for this to be true, one would have to assumethat the field variables on quasi-de Sitter spacetime can exist as an effective field theory onscales smaller than (cid:96) Pl , which is manifestly problematic from our understanding of quan-tum gravity. Note that the pinnacle of success of inflation lies in explaining macroscopicperturbations, which source the structure formation of the universe, as originating fromquantum vacuum fluctuations. From this point of view, the TCC simply turns around thiscrowning glory of inflation to posit that any accelerated phase of expansion can only bevalid for a finite amount of time and not be semi-infinite in the past. The upper limiton the duration of this accelerating background is set by requiring that any mode, with aphysical wavelength equal to or smaller than (cid:96) Pl , should never cross the Hubble horizon(H − ) so that it does not decohere and become part of the classical perturbations.On the contrary, an immediate obstruction to such a way of thinking comes from thefollowing – as one traces back a classical perturbation mode, linear perturbation theorybreaks down much before the physical energy, corresponding to the given mode, becomesof the order of M Pl , the four-dimensional Planck mass . In other words, even before thephysical wavelength of a given perturbation mode can get to O ( (cid:96) Pl ), the linear perturbationwould become comparable to the magnitude of the background variable, thereby breakingdown perturbation theory. This conclusion makes sense if one considers the fact that theuniverse is extremely inhomogeneous and anisotropic on Planck scales and, therefore, onecannot use the approximation of linear perturbation theory and quantize fluctuations interms of its Fourier modes. Indeed, as long as one considers the expanding backgroundwithin an effective field theory description of gravity, it must break down on wavelengthssmaller than (cid:96) Pl since such energies would collapse parts of space-time into black holes(or a collection of them) [64]. Therefore, from this point of view, it might indeed seemconceivable that the argument presented above for the trans-Planckian problem shouldnever arise since it stretches the effective field theory of linear perturbations beyond itsrealm of validity.However, one must keep in mind that the above heuristic idea is not the main theoret-ical argument behind the TCC and should rather be viewed as an intuitive understandingof the problem. The principle conceptual difficulty is that of non-unitarity of the Hilbertspace of the perturbations [25, 65], as can be understood as follows. Recall that one needsto impose a UV cut-off even for quantum field theory on flat spacetime, for the purposes ofrenormalization, to get physically meaningful answers. In the case of gravity, the UV cut-offis not just a computation tool but is rather a physical one, given by M Pl . In analogy withMinkowski spacetime, one would then expect that imposing such a cutoff would get rid ofthe trans-Planckian problem and give us a well-defined, decoupled effective field theory ofinflation below scales of O (M Pl ). However, this is precisely where the main difficulty asso-ciated with expanding backgrounds show up. The problem is that for such spacetimes, theUV cut-off must be fixed in physical coordinates while the Fourier modes have wavelengths We symbolize this differently from M p which was used to denote the eleven-dimensional Planck mass. – 99 –hich are expanding in those coordinates. Therefore, a mode whose physical wavelength isabove M Pl to begin with, during inflation, might have its wavelength red-shifted to energiesbelow M Pl and thus would be part of the low-energy effective field theory. In this way,more and more modes would redshift from the UV into the Hilbert space of system modesdescribing perturbations on top of an expanding background and make it time-dependent.A time-dependent Hilbert space, having to accommodate more degrees of freedom to ex-plain physical phenomena with time-evolution, is a classic sign of non-unitarity creepinginto the theory. Of course, this is a problem associated with any expanding background.What is special for (quasi-)de Sitter setups is that some of these UV modes can eventuallycross the Hubble radius and thus become observable at late times. From this point of view,one can view the TCC as a requirement of keeping this non-unitarity hidden behind theHubble horizon so that even if these trans-Planckian modes somehow get generated, theynever become part of our low-energy system.Having set up the trans-Planckian problem elaborately, let us now explain how oursolution manages to evade it. First, let us give some estimates for the relevant time-scalesinvolved in the problem. The TCC postulates that any trans-Planckian mode should nevercross the Hubble radius so that it cannot decohere and become classical. This can bemathematically formulated as an upper bound on inflation, given by [24] N < log (cid:18) M Pl H f (cid:19) , (4.1)where N is the number of e -foldings of inflation and H f is the value of the Hubble parameterat the end of inflation. For a meta-stable de Sitter spacetime, this translates into an upperbound for the lifetime of such a solution, given by [24] : T <
1H log (cid:18) M Pl H (cid:19) . (4.2)Thus, according to the TCC, the lifetime of any metastable de Sitter spacetime should bebounded as above for it to avoid the trans-Planckian problem. On the other hand, severalarguments for assigning a finite entropy to de Sitter spacetimes leads to a bound of theform [66]: T < S dS = 1H (cid:18) M Pl H (cid:19) . (4.3)A similar bound was also derived by treating de Sitter as a coherent state on top of aMinkowski vacuum in a toy model, the upper limit coming from the ‘quantum break-time’ of the system, after which the interaction terms lead to the breaking of the semi-classical description of the system [29]. There has been a fierce debate recently as towhich of these two time-scales should be treated as the maximum allowable lifetime ofa consistent metastable de Sitter vacuum [67]. For completeness, let us point out thatthe crucial argument that quantum modes become classical after they cross the Hubble H denotes the Hubble parameter in this section, and should not be mixed with the warp-factor, H ( y ),from the earlier sections. – 100 –orizon has indeed been challenged in [68], pointing out mechanisms (such as that ofparametric resonance) which can ‘classicalize’ it even within the Hubble radius. Moreover,several arguments from string theory, such as the distance conjecture or the weak gravityconjecture, also gives rise to a refined version of the TCC [69] with an O (1) numberappearing on the RHS of (4.1) and (4.2). In light of this, what can be unambiguouslystated is that there is a time-scale beyond which any consistent description of a de Sitterspacetime should break down, albeit the upper limit on the lifetime is still under contention.However, note that in our case the amount of time we can trust our de Sitter solutionas described by a Sudarshan-Glauber state, before the system becomes strongly-coupled, isgiven by | T | < / H. As shown in (2.80), after this time, the string coupling becomes g s ∼ vacua which appear in string theory typically have amuch longer lifetime [24, 70], such as in the KKLT [5] and LVS scenarios [71] .Although the above heuristic arguments are interesting, let us now come to the real rea-son why our de Sitter solution remains unscathed by the above-mentioned trans-Plackianproblem and, therefore, the TCC. The crucial underlying reason is precisely the fact that itis not a vacuum solution but rather a coherent state on top of a (warped-) Minkowski vac-uum. To understand this better, let us revert to the effective field theory (EFT) approachto the trans-Planckian problem of inflation. Typically, for an EFT on a flat spacetime,one would expect that effects of trans-Planckian physics would be suppressed by factorsof O (cid:0) H / M (cid:1) [67]. However, the non-unitarity associated with expanding backgroundsmanifests itself in a way such that there are parts of parameter space in an EFT of inflation,in which there are trans-Planckian effects larger than this, namely violating expectationsof de-coupling of inflation from Planck scale physics [72]. So how does decoupling work forperturbations in inflation? Indeed, it is known that if one makes the following assumptions[25]:1. The microscopic structure of space-time, on Planck scales, is Lorentz invariant, and2. The perturbations (or the expansion of the field modes on the de Sitter spacetime)are in their local vacuum,only then does the effects of decoupling kick in and the trans-Planckian problem mentionedabove goes away. In this case, one can show that the probability of producing a trans-Planckian mode in the theory is exponentially suppressed, given by e − M / H , due to aBoltzmann factor. Although the above-mentioned assumptions seem very strong for aclassical de-Sitter spacetime, say, with a Bunch-Davies vacuum, it is exactly what we havein our construction of de Sitter as a Glauber-Sudharshan state. Our solitonic vacuum isindeed supersymmetric, warped-Minkowski and satisfies both of the above criteria. The reason for this is that we do not depend on some gravitational decay channel such as through theColeman-de Luccia tunneling. Rather, it is the system becoming strongly-coupled that determines the timefor which we can trust our solution. – 101 –t this point, the acute reader might ask how does our de Sitter solution solve theunitarity problem mentioned above? As was clear from the discussion above, the trans-Planckian difficulties arise as an effect of having time-dependent frequencies associatedwith the perturbation modes. However, as was manifestly shown earlier in (2.87), thetime-dependencies of the frequencies of perturbations in our case are actually artifacts ofFourier transforms over a de Sitter state, viewed as a Glauber-Sudarshan state. Anotherway of seeing that the trans-Planckian problem cannot arise in our paradigm is due to thefact that the perturbations on top of our Glauber-Sudarshan state can be expressed as anAggarwal-Tara state over the same Minkowski vacuum . In other words, the dS pertur-bations with time-dependent frequencies can be rewritten as (infinite) linear combinationof perturbations, with time-independent ones, as shown in (2.25) and (2.87). This is thecrucial reason why we are able to write down a well-defined Wilsonian effective action andnever have to resort to the TCC for solving the puzzle of the trans-Planckian modes asthey are necessarily decoupled, as they should be for the low-energy effective action. All ofthis also fits in nicely with the intuitive expectation that the trans-Planckian problem findsits resolution in a UV-complete theory of inflation, which is the case for our system. Ourwarped-Minkowski vacuum, on which the de Sitter coherent state is constructed, arisesin string theory, thereby resolving these trans-Planckian problems without having to hidethem behind the Hubble horizon as was proposed by the TCC.
There is a different point of view proposed as to why de Sitter solutions might indeed bein a swampland, heuristically relating the instabilities of de Sitter spacetime, coming fromfield theoretic arguments, to the de Sitter conjecture [40]. The basic idea is that evenif one is able to find an effective potential which gives rise to a de Sitter vacua within astringy construction, at the classical level, then radiative loop corrections would necessarilydestroy it leading one into a quantum version of the swampland. It is so because even forclassical solutions which obey the equation of state p = − ρ , the one-loop effective potentialessentially breaks this, yielding w (cid:54) = 1, unless one assumes the Bunch-Davies vacuum forthe fields on top of the de Sitter spacetime. However, it has been argued that the Bunch-Davies is a rather unnatural choice for the vacuum [73] and thus should be discarded. Onthe other hand, any other sensible choice of the vacuum necessarily leads to the leaking ofthe cosmological constant, and one gets into the quantum swampland.At first sight, one might wonder if the choice of the quantum vacuum should playany serious role in the search of de Sitter vacua in string theory. After all, in the absenceof full-fledged string loop calculations, i.e. quantum correction in spacetime, as can onlyarise in string field theory, the main focus has been to derive effective potentials whichcan support a de Sitter solution using stringy effects. In fact, one might even wonder if aclassical (or non-perturbative) de Sitter vacua which is ruled out by the swampland caneven be resurrected by employing these radiative loop corrections. However, as has beenshown in [40], these quantum loop corrections cannot improve the stability of the solution(although they can affect the value of the cosmological constant). More interestingly, only– 102 –xamining quantum field theoretic calculations on de Sitter spacetimes, one can establisha relation between them and the swampland.The crucial realization behind the argument for the quantum swampland is the non-uniqueness of the vacua for de Sitter space. Typically, one chooses the Bunch-Daviesvacuum by expanding a field in its momentum modes, picking one of these modes andblue-shifting it backwards until the effects of de Sitter spacetime can be ignored. At thispoint, one can safely pick the unique Minkowski vacuum. This procedure can be repeatedfor all the momentum modes to arrive at the Bunch-Davies, or the Euclidean, vacuum.However, as already mentioned in our discussion on the TCC, this procedure cannot workif there is a fundamental UV cutoff since one cannot trace a given mode beyond this energyscale. Moreover, the Bunch-Davies is not the only de Sitter-invariant vacuum; rather, thereis a whole family of vacua which respects the symmetries of de Sitter spacetime called the α -vacua [74]. There are also complementary ways of showing the rather fine-tuned andcontrived nature of the Bunch-Davies vacuum, especially when considering a causal patchof de Sitter instead of global de Sitter, as is appropriate for our solution [73]. Havingargued that the naive choice of the Bunch-Davies vacuum is a probably a wrong one, [40]shows that as soon as one chooses a different vacuum (such as an instantaneous Minkowskivacuum [75]), the cosmological constant leaks and leads to a quantum decay of the de Sitterspacetime.Let us now understand how our de Sitter solution can never run into these problemsassociated with radiative corrections. As shown explicitly, the Bunch-Davies has a verydifferent interpretation in our construction as a (generalized) Agarwal-Tara state. Our deSitter is itself a built as a coherent state on top of a solitonic Minkowski solution, andperturbations on top it is viewed as a GACS on top of a Minkowski state. Firstly, choosingdifferent coefficients in the definition of our GACS would lead to a different vacuum statefor the fluctuation modes. Just as it was shown how one can reproduce the Bunch-Daviesstate starting from our generalized Agarwal-Tara state (2.86), we can also reproduce otherde Sitter-invariant vacua, such as the α -vacua, for different choices of the C ( ψ ) n k ( t ) in (2.47).However, remember that for us it is always the solitonic background, with the correspondinginteracting vacuum | Ω ( t ) (cid:105) , on which we build both our Glauber-Sudarshan state and theAgarwal-Tara state for fluctuations on top it. Therefore, loop corrections do not spoilour solution since these radiative effects are all calculated with respect to a Minkowskibackground and not a de Sitter one. More to the point, our de Sitter solution is constructedonce we build the Glauber-Sudarshan state having taken all types of quantum corrections– perturbative and non-perturbative, local and nonlocal – into account. Essentially, we donot build an effective vacuum spacetime with an equation of state w = − H int , the description which is validfor a specific amount of time. The radiative corrections having been calculated for ourinteracting vacuum on flat spacetime do not lead to the same pathologies as they do forde Sitter space. In fact, the main argument for the quantum swampland was based on theambiguity of choosing the vacuum in de Sitter space. However, in our case, there is only– 103 –ne clear vacuum in our theory – the interacting vacuum | Ω (cid:105) due to the action of H int onour solitonic vacuum – and we build both our Glauber-Sudarshan and Agarwal-Tara stateson top of this. Consequently, the loop corrections do not affect the stability of our solutionas long as g s (cid:28) In spite of the Gibbons-Hawking entropy being a natural extension of the Bekenstein-Hawking entropy associated with a black hole (locally, the horizon of de Sitter is identicalto that of a Schwarzschild black hole), the finiteness of the entropy of de Sitter space hasbeen a long-standing puzzle. Since black holes are local objects, occupying a finite regionof space, it is natural to finite entropy to a black hole which, in turn, implies that the afinite number of states can describe the black hole system. On the other hand, the spatiallyflat slices of de Sitter space has infinite volume. And yet, as demonstrated by Gibbons andHawking through Euclidean partition functions [76], an inertial observer in de Sitter spacedetects thermal radiation at the temperature: T dS = 12 π(cid:96) , (4.4)where, (cid:96) ∼ / Λ is the length scale related to the cosmological constant. One can thenuse the first law of thermodynamics to deduce that the entropy corresponding to de Sitterhorizon, from the Gibbons-Hawking temperature , to be: S dS = A G ∼ π(cid:96) G , (4.5)with the horizon area A ∼ π(cid:96) , with G is the Newton’s constant. The finiteness of thede Sitter entropy stands out as a crucial test for any quantum gravity theory. Beforedescribing how our description of de Sitter as a Glauber-Sudarshan state in string theorymanages to explain the finite entropy of the resulting de Sitter spacetime, let us quicklyreview some of the known features of S dS .Firstly, it is clear from the above discussion that the finiteness of entropy must somehowbe related to the fact that any single observer has access to only a finite volume of deSitter space. Note that this reference to an observer is crucial for the discussion of deSitter space unlike in the case of black holes. We emphasize that the entropy is only finitesince the cosmological horizon ensures that a given observer only can ever send signals toa finite portion of the universe. From this simple observation, one can draw the minimal The first law can be expressed as ( ∂S/∂M ) = T − ; however, a priori, there is no definition of themass corresponding to the de Sitter horizon. The way out of this is to realize that we need only a massdifferential and this was solved by introducing a negative mass. We gloss over these subtleties as they arequite well-known and have been discussed exhaustively in the literature (see [77] for an overview). – 104 –onclusion that any effective field theory breaks down when one has e S dS states behind thehorizon. Indeed, if one turns off gravity by taking G →
0, while still maintaining the samecurved space geometry, the entropy does go to infinity and one has a perfectly valid EFTdescription [66].Next, we need to understand the serious consequences one has for the underlyingquantum gravity theory if indeed S dS is to be finite. It has been noted that eternal deSitter space has several conflicts with having a finite entropy. In [78], a thermofield doublepicture was developed to explain the finite thermal entropy of a causal patch of de Sitter,using arguments from complementarity. However, the main conclusion of this work was toshow that the symmetries of de Sitter spacetime were incompatible with the finiteness of S dS . More explicitly, it was found that the Hamiltonian (as a generator of the de Sittersymmetry group) can only have a countable spectrum, which is required to have discreteenergy eigenvalues and a finite entropy, if the symmetries are violated on time-scales inwhich this discreteness become significant. The relevant time-scale in this case would bethe Poincar´e recurrence time, t p ∼ e S dS , but the important point for us is that this providesa concrete argument against the existence of eternal de Sitter spacetimes coming from thefiniteness of S dS . This is quite a remarkable finding since, classically, one only requires thatde Sitter is the solution of Einstein’s equations with a positive cosmological constant andcan be eternal in the future and yet, this is ruled out once the finiteness of S dS , itself asemiclassical result, is taken into consideration.A stronger requirement for quantum gravity, due to the finiteness of S dS , would be therestriction that the Hilbert space is finite dimensional [79] and given by N ∼ e S dS , where N is the number of states on the Hilbert space, once the covariant entropy bound is takeninto account. In other words, ruling out entropies larger than S dS imposes a fundamentalcutoff on the Hilbert space of the quantum gravity theory. This is the so-called Λ − N correspondence [80], with the number of degrees of freedom, N , being given by N = log N , N being the number of states on the Hilbert space. A small but finite Λ, necessary fora finite S dS , ensures that the fundamental theory would have a very large, and yet finite,number of degrees of freedom. However, it should be emphasized that this interesting,and strong restriction, relating the size of the Hilbert space with the cosmological constantrequires additional conditions (such as a future asymptotic de Sitter region [81]) . A finalpoint to emphasize, which has already been mentioned earlier and is extremely relevantfor our discussion, is that there exists other arguments which show that the finiteness ofentropy results in an upper limit on the lifetime of the de Sitter space [66].From our review of topics above, it should be clear to the reader that we want to firstfocus on the fact that our description of de Sitter, as a Glauber-Sudharshan state, can onlybe trusted for a finite amount of time given by T < / H (2.80). Therefore, our solutionautomatically satisfies, at least, the necessary criterion of [78] for having a finite entropy,as this time-scale is much smaller than the Poincar´e recurrence time t p . As an aside, let us The Friedmann equation gives a constant Hubble parameter 3M H = Λ for an eternal de Sitter space,sourced by a constant positive cosmological constant. Note that having a positive Λ is not sufficient to guarantee the above conclusion due to the failure ofhaving a covariant entropy bound in some cases. – 105 –lso mention that our solution is completely free from problems such as that of Boltzmannbrains (see [82] for details). This is so because the time-scale, characteristic of Boltzmannbrains, is given by: T BB = 1H e S E , (4.6)where S E is the instantonic action corresponding to a Boltzmann brain. However, given thetime-limit after which our system becomes strongly-coupled, this would mean S E ≤ S dS , we are yet to specify the microscopic mechanism through whichsuch an entropy is generated. Note that it is a major challenge of any fundamental theoryto calculate statistical entropy of de Sitter space (See, for instance, [83] for pioneering workin this direction for lower dimensional de Sitter space). In our case, we wish to interpretthe entropy of the resulting de Sitter spacetime as an entanglement entropy due to theinteraction between the modes of the metric (and G-flux) fluctuations which give rise tothe Glauber-Sudarshan state on top of the solitonic vacuum. Although we shall not givethe detailed calculation, which we defer to future work, we can nevertheless point out whywe expect such an entanglement entropy to exist and, moreover, why it should be finite.Let us begin with the latter point first. The number of gravitons in a given coherentstate is, of course, infinite once we allow for modes with all possible momenta. This isexplicitly shown in (2.51) for our Glauber-Sudarshan state. However, as also mentionedearlier, one must have a short-distance cut-off in order to have a well-defined Wilsonianeffective action. Indeed, one can also impose a physically-relevant infrared cut-off in thecase of the coherent state giving rise to de Sitter. However, let us come to this point later.The first question is what does a finite number of gravitons in the Glauber-Sudarshan statehave to do with a finite entropy?Before we answer this question, note that we are not the first to show a relationbetween a coherent state description of the causal patch of de Sitter and it having afinite entropy. In [29], a coherent state constituting of soft gravitons was proposed asthe “quantum-corpuscular” description of de Sitter space. Although not formulated fromany fundamental theory (such as string theory), this still leads to a time-limit (4.3) afterwhich such a semiclassical description of de Sitter stops being valid, as mentioned earlier.The number of gravitons in such a state can be found by using the number operator: N = (cid:104) N | ˆ N | N (cid:105) , where the coherent state | N (cid:105) = Π k | N ( k ) (cid:105) should be calculated over allfrequencies: | N ( k ) (cid:105) = exp (cid:18) − N ( k )2 (cid:19) ∞ (cid:88) n k =0 [ N ( k )] n k / √ n k ! | n k (cid:105) . (4.7) Of course, we should also mention G-flux particles but, in order to keep the discussion less complicated,we shall only focus on the part of the metric fluctuations of the Glauber-Sudarshan state ( | α (cid:105) in ournotation) instead of focussing on the full coherent state | α, β (cid:105) . Note that this is done only for convenienceand the discussion easily generalizes to the full case even if explicit computations become more tortuous inthat case. – 106 –his would also, naturally, be an infinite number but the authors of [29] make the crucialassumption that constituent gravitons of the coherent state satisfy the condition that thedominant wavelength is the one set by the Hubble radius H − ! This leads to a gravitationoccupation number given by, N ∼ M / Λ, which coincides numerically with the de Sitterentropy S dS . First, notice that the finite occupation number of this coherent state picturecomes from the requirement that de Sitter is a good semiclassical, mean-field descriptiononly as long as the occupation number is related to the cosmological constant as givenabove. On the other hand, our Glauber-Sudarshan state, describing de Sitter in full stringtheory, has a finite occupation number since we require to have a short-distance cutoff. Thisis an essential difference and we do not assume that only gravitons of a specific frequencycontribute to the coherent state.Given a coherent state with a finite number of gravitons, N , one can calculate howmany states can correspond to having such a description. Of course, if these gravitonswere purely non-interacting, then the number of states would be given by N γ , γ being thenumber of states (or polarization) of individual gravitons. However, it was heuristicallyargued in [29], that if the gravitons do interact, then the number of states would go as ξ N ,where ξ denotes the number of states of individual distinguishable “flavors” of gravitons,the latter being an effect of interaction between gravitons. This would give a leading orderentropy given as S ∼ N , with logarithmic corrections. We refer the interested reader to[29] for details of this estimate.Although we agree with the general argument of having interacting gravitons, let usshow how it leads to a finite de Sitter entropy in a more rigorous way. Recall that the totalnumber of gravitons in our Glauber-Sudarshan state is given by (2.51): N ( ψ ) = (cid:90) Λ UV d k (cid:12)(cid:12)(cid:12) α ( ψ ) k (0) (cid:12)(cid:12)(cid:12) , (4.8)where we have explicitly introduced a short-distance UV cut-off Λ UV to replace the ∞ appearing in (2.51). Firstly, as mentioned earlier, this number is finite in our constructionby virtue of the fact that the effective action must come out of integrating the high-energyUV modes and not by requiring that the coherent state is packed with gravitons of aspecific wavelength. However, in addition to the UV cut-off, an interesting question nowarises whether there is any IR cut-off for us. Recall that our fluctuations, underlying thestate | α (cid:105) which give rise to de Sitter space, are over a warped-Minkowski vacuum and wedo not have any inherent preference for the IR cut-off. Nevertheless, we are interested incalculating the entropy corresponding to a causal patch of de Sitter whereas our solutionrepresents the full de Sitter spacetime in the so-called flat slicing (see Figure 1 ). Therefore,it is natural to identify the IR cut-off with the Hubble scale, i.e. Λ IR = H. However, thereis no way to “integrate” out the IR degrees of freedom to get an effective action for modesin-between Λ UV < k < Λ IR . It is well-known that in this case, the standard treatmentrequires the description of the modes of interest in terms of the density matrix obtainedafter tracing out the IR modes. As mentioned, we shall sketch the outline of this calculationbelow, following the notation of [84], while deferring the details to later work.– 107 – igure 1: The figure on the left shows the Penrose diagram for global four-dimensional de Sitterspace. The poles O N and O S are time-like lines. The dashed lines denote the past and futurehorizons for an inertial observer at O S . Time runs from the past J − to the future J + conformalboundaries.On the right, we have the Penrose diagram for the flat slicing of de Sitter, as is applicable for oursolution. The causal patch of an observer, sitting at O S , is denoted by the shaded region (staticpatch). We trace over the modes in Region III to get the entropy corresponding to region I. TheHubble horizon separates regions III from I. For our quantum system, we break up the effective Hamiltonian, corresponding to theWilsonian effective action, into the following form: H = H sys ⊗ + ⊗ H IR + H int , (4.9)where we have denoted the H sys as the Hamiltonian corresponding to the perturbationsmodes Λ UV < k < Λ IR while H sys refers to modes with k < Λ IR . The interaction Hamil-tonian is the same as the one introduced in (2.55). Effectively, one breaks up the Hilbertspace of all the modes which make up the de Sitter coherent state into H = H sys ⊗ H IR .The crucial point for us is that these modes, although complicated due to the underlyingsolitonic vacuum, still has a time-dependence of the form e iω k t . Of course, this was im-portant for us to begin with in the construction of the Glauber-Sudarshan state but here,its significance lies in the fact that we shall be able to use time-independent perturbationtheory because of this. We shall see this shortly.Let us first denote the free vacuum state of the theory, before considering the H int as: | , (cid:105) = | (cid:105) sys ⊗ | (cid:105) IR , (4.10) i.e. as a (tensor) product of the individual vacuum states. However, once we turn oninteractions, the only vacuum available to us is the interacting vacuum given in (2.55),as we have emphasized many times. Starting with this interacting vacuum, which can be– 108 –ritten as: | Ω (cid:105) = | , (cid:105) + (cid:88) n (cid:54) =0 A n | n, (cid:105) + (cid:88) N (cid:54) =0 B N | , N (cid:105) + (cid:88) n,N (cid:54) =0 C n,N | n, N (cid:105) , (4.11)we want to trace out the IR modes. In the above, we have denoted energy eigenstates,corresponding to system and IR modes, by | n (cid:105) and | N (cid:105) , respectively. This is where we shalluse perturbation-theory to calculate the co-efficients A, B, C . In fact, the density matrixcorresponding to the system modes, can be written in terms of the matric elements of C alone (up to leading order) [84]: ρ sys = tr IR | Ω (cid:105) (cid:104) Ω | = (cid:32) − | C | CC † . (cid:33) (4.12) C can be formally expressed as C n,N = |(cid:104) n, N | H int | , (cid:105)| E + ˜ E − E n − ˜ E N + · · · (4.13)where the · · · above refer to higher order corrections in O ( g s ). Given this reduced densitymatrix ρ sys , we can calculate the quantum von Neumann entropy corresponding to it, givenby S ent = − tr ( ρ sys log ρ sys ) , (4.14)in terms of the matrix elements given above. The above result holds true for arbitrarydimension. As mentioned, we do not wish to do this explicit calculation here which wouldnot only involve the H int , including all the quantum terms described in (3.63), but alsorequire including the full Glauber-Sudarshan state corresponding to both metric and G-fluxfluctuations. Nevertheless, our main result can be understood as follows.Firstly, we note that we give a precise microscopic origin to the de Sitter entropy inour formalism by relating it to the quantum entanglement entropy of the mode functions.We stress that this is not the usual entanglement entropy one sometimes calculate forfields on de Sitter space [85] but rather that of the modes which give rise to the de Sitterspacetime itself. The main reason why we are able to do such an identification is simplybecause our de Sitter space comes into existence on considering fluctuations of the metric(and G-flux components) over a solitonic vacuum. The entanglement entropy is the en-tropy corresponding to the coupling between these modes themselves, which build up theGlauber-Sudarshan state itself. In other words, the interactions between the gravitons andflux-particles, which constitute our de Sitter coherent state | α, β (cid:105) , is responsible for theorigin of this entanglement entropy and it is thus natural to associate it with S dS . How-ever, note that entanglement is purely a quantum property and thus we give a statisticalexplanation of S dS in our formalism for a de Sitter spacetime in string theory.Secondly, the inquisitive reader might question our sketch of the standard derivationabove, given that we have used | Ω (cid:105) to compute the entanglement entropy, and not theGlauber-Sudarshan state corresponding to it D ( α ( t )) | Ω (cid:105) , in (4.11), before tracing out the– 109 –R modes. This is a very pertinent point; however, the simple calculation involving thevacuum suffices in this case as it has been shown that the entanglement entropy corre-sponding to any coherent state is exactly the same as that for the vacuum state [86], aconclusion that lends itself to any dimensional spacetime.Next, we arrive at the question of the finiteness of the entanglement entropy. This is thekey property which would allow us to identify it with S dS . Note that in the absence of anyinteractions, i.e. setting H int = 0 would result in the entanglement entropy going to infinity,as is expected for a free field theory. On the other hand, given a finite interaction term, andthe fact that we have a UV cut-off Λ UV , ensures us that this quantity remains finite. Inour case, the full interacting action of M-theory was meticulously spelt out in the previoussection and, in fact, it was emphasized at the very outset of our construction of the Glauber-Sudarshan state that interactions were absolutely crucial for such a description to emerge.There is simply no free underlying theory available to us when we consider G-fluxes andobtain our solitonic background. If we turn off interactions, no construction of a de Sitterspacetime is possible and thus the entanglement entropy would simply have no meaning ofbeing associated with S dS . What we stress is that we need not invoke any physical intuition,such as that of the coherent state being built out of gravitons of any specific wavelength,in order to obtain a finite entropy of de Sitter space. Furthermore, the intuition thatinteractions between the graviton constituents lead to the entropy corresponding to S dS was rigorously shown to be associated with the entanglement entropy arising due to thegraviton mode-couplings between those of the causal patch and those which are traced outin the far-infrared.Finally, we need to justify our choice of tracing over the IR modes in order to getour entanglement entropy. This brings us back to something which we mentioned at thevery beginning of this section: The finiteness of entropy is related to the fact that anobserver in the static patch only has access to a finite part of de Sitter space. In otherwords, for any inertial observer, the entropy corresponds to the region of de Sitter hiddenbehind the cosmological horizon. Therefore, it makes perfect sense for us to trace out overmodes, which correspond to the region that is not in causal contact with the observer.What determines this region, and therefore the IR cut-off, is the cosmological constantwhich, in our case, is emergent as a balance between fluxes and quantum corrections [21].We emphasize that the tracing out carried out over here is not the usual one used forthermofield-double systems – starting with some (Hartle-Hawking type) Euclidean statewhich is entangled between left and right static patches and then tracing over the hiddenregion to obtain a thermal density matrix [78]. Rather, our state is always the Glauber-Sudarshan state created over the interacting vacuum, for which we trace over the modeswhich have wavelengths larger than our IR cut-off. The physical reason behind choosingthe cutoff is that, in the resulting de Sitter space, an observer in the static patch can onlycausally interact with the region that is not hidden behind the horizon.Of course, we did not do the explicit calculation for the entanglement entropy takingthe full interacting Lagrangian of M-theory into consideration, as well as carrying out theactual trace over the IR degrees of freedom. However, one should be convinced by nowthat such a procedure is, in principle, possible due to the time-dependence of our mode– 110 –unctions Ψ k ( x , y, z, t ) even if their algebraic forms are complicated due to the solitonicbackground. Furthermore, the result of such a calculation would give us a finite answer.What would then be left is to equate this result to the de Sitter entropy S dS (as one-quarter the horizon area) to find what it predicts for the allowed wavelength distributionof the gravitons comprising our Glauber-Sudarshan state. There is one final hiccup in ourargument which shows up in the form of a well-known problem against interpreting S dS asan entanglement entropy, since the latter S ent would not only depend on the cutoff Λ UV butalso on the number of species. This latter dependence is what is known as the species puzzle[87]. However, if one introduces the species bound [88], then the cut-off which shall appearin the calculation of S ent shall exactly be this effective cutoff, as demanded by the speciesbound. As goes the usual argument, if one has a length scale cutoff (cid:96) eff > √ N (cid:96) Pl in theresulting theory, it is natural that this would appear as the cutoff for the wavelength of thegravitons populating our Glauber-Sudarshan state. However, a pertinent question wouldbe if there are a lot of light species present in our formalism such that this effective cutoffis reduced by an unacceptably huge amount? And this is precisely where our dynamicalmoduli stabilization comes to the rescue. As explained earlier, this means that the moduliare fixed at every instant of time in such a way that, at every instant of time, the Dine-Seiberg runaway is stopped. This ensures that we do not have exponentially light statesappearing in our setup at any time and thus manage to avoid the species puzzle.Finally, note that the entropy corresponding to the reduced density matrix in (4.12)would be the leading order result, with higher order corrections in g s to follow. Thus, aftersetting this leading order result to S dS , we shall also be able to systematically calculate thehigher order corrections to the semiclassical result. This is as it should be for any trulymicroscopic understanding of S dS and is only possible since we have a UV complete theory– string theory – describing our background.
5. Discussions and conclusions
In this work we investigated the realization of de Sitter space from string theory. Therewere many attempts to perform such constructions with various degrees of success, thatincluded ingredients like fluxes, non-perturbative effects from instantons, orientifolds andanti-branes. Our approach differs from all the previous attempts as we consider the ap-pearance of the four-dimensional de Sitter metric from a Glauber-Sudarshan (coherent)state in string theory. The foundation of our construction is four dimensional Minkowskivacuum obtained from string theory or M-theory compactifications as in (2.1). The latterrealization, i.e. the uplift to M-theory, is only for convenience as it aids in making some ofthe computations easier to perform. The vacuum, either in IIB or in M-theory, is super-symmetric and stable and is well understood even in the presence of quantum corrections.These corrections, while necessary to realize the supersymmetric background itself, convertthe vacuum from a free to an interacting one. This interacting vacuum forms the basis ofour subsequent constructions in the paper. For example, instead of switching to anothervacuum which has the potential to be a non-supersymmetric de Sitter, we realize our de– 111 –itter space by using a displacement operator on the interacting vacuum itself. One of ourmain result of the paper is the precise identification of the displacement operator.Displacing the interacting vacuum appropriately creates the necessary coherent orthe Glauber-Sudarshan state from which one could extract the precise four-dimensionalde Sitter metric − along-with the metric information of the internal six-manifold − bytaking the expectation values of the various components of the metric operator over theGlauber-Sudarshan state. Additionally, it provides the information of the fluxes that arerequired to support the background metric configuration, either in the IIB or in the M-theory side. While the fluxes are supersymmetric over the Minkowski vacuum, they breaksupersymmetry when we take the expectation values. A precise demonstration of thesefacts forms the basis of section 2 of the paper.Another pleasant surprise of our approach is that not only the de Sitter space arisesform the Glauber-Sudarshan state but also do the fluctuations over the de Sitter space.As an extension of electromagnetism inspired concepts, the fluctuations over a de Sitterspace appear as a generalized Agarwal-Tara state where the photon-added coherent stateis promoted to a generalized graviton (and flux)-added coherent state. This state is alsoidentified for various components of the metric, details of which are shown in sections 2.3,2.4 and 2.5.Many of the computations, demonstrating the aforementioned details, are performedin two ways throughout the paper: one, using state and operators and their expectationvalues; and two, using the path integral approach. We show that the latter approach ismuch more powerful and ties up with many well-known concepts in string theory, like theaction of the vertex operators, flux induced non-K¨ahlerity etc. Additionally, some of thelimitations of the state-operator formalism, for example being restricted to mostly on-shellcomputations or the appearance of un-necessary divergences, are effectively eliminated inthe path integral approach.The path integral approach also allowed us to choose between the three possible out-comes of our analysis, namely: (a) retainment of the exact classical behavior, as expectationvalues, over an indefinite period of time during the temporal evolution of the system, (b)persistence of the exact classical behavior for a certain interval of time beyond which thedominance of the full quantum behavior becomes prominent, or (c) appearance of theperturbative corrections to the expectation values at least in some well defined temporaldomain debarring the system to exhibit the full classical behavior anywhere in the domain.This temporal domain, which is sometimes referred to as the quantum break time , is animportant limitation of the system. In our case it is governed by the interval beyond whichstrong coupling effect sets in (at least from M-theory point of view which we use to analyzethe dynamics) and we lose quantitative control of the dynamics, thus effectively eliminatingoption (a) above. It is therefore option (b) that eventually appears consistent from ourpath integral approach, as any perturbative corrections appearing from option (c) wouldhave implied time-dependent Newton’s constant, non-exactness of the four-dimensional deSitter solution, and other possible pathologies. A precise demonstration of these appear insections 2.4 and 2.5. An important result of these sections is that the Agarwal-Tara stateaccurately reproduces the fluctuations over the Glauber-Sudarshan de Sitter space. The– 112 –ame state also allows an interpretation of the trans-Planckian issue, as resulting from thetime dependent frequencies that we observe in mode expansions of the fluctuations overa de Sitter vacuum, to be simply an artifact of the Fourier transforms over the de Sitterspace viewed as a Glauber-Sudarshan state. This shift in the viewpoint of de Sitter frombeing a vacuum to a state is crucial in resolving the trans-Planckian problem and therebyallowing Wilsonian effective action to be defined at all energy scales.Stability of the Glauber-Sudarshan is also an important criterion on which all of ourconstruction relies on. The analysis of the stability requires two different sets of compu-tations, one, analyzing all possible perturbative and non-perturbative corrections affectingthe system, and two, analyzing the equations that govern the dynamical evolutions of theexpectation values of the metric and the flux components over the Glauber-Sudarshanstate. The perturbative corrections have been discussed in details in [21, 22] and in section3.1 and 3.2 we elaborate on all possible non-perturbative corrections, including the onesfrom the instantons, and the world-volume fermions on the seven-branes.The equations governing the dynamical evolutions of the expectation values are theSchwinger-Dyson’s equations (SDEs). Interestingly, the SDEs reproduce all the M-theoryEOMs in the presence of the aforementioned perturbative and the non-perturbative cor-rections. These equations may be classified order by order in type IIA string coupling g s (which becomes a time-dependent quantity) with the lowest order equations determiningthe de Sitter background and the fluxes supporting it. The stability then works in thefollowing way. The higher order flux components and the higher order metric components,balance against the higher order quantum terms to keep the lowest order SDEs unchanged.This balancing act happens to all orders in g s and M p such that the de Sitter backgroundalong-with the supporting flux components remain uncorrected to arbitrary orders in g as M bp .Such a stability criterion also guarantees option (b) mentioned above, namely the domi-nance of the exact classical behavior in the temporal domain whose boundary is dictatedby the onset of type IIA strong coupling. The moduli are stabilized already at the vacuumlevel, and therefore the dynamical evolution of the metric components also govern the dy-namical evolution of the moduli themselves disallowing, in turn, the Dine-Seiberg runawayat every stage of the evolution as long as we restrict the dynamics within the allocatedtemporal domain.Having explained how our solution is able to go past the technical difficulties whichhave been pointed out for realizing four-dimensional de Sitter space in string theory, wewent on to explore some of the properties of the constructed Glauber-Sudarshan state insection 4. The first thing we did was to show how this construction is able to bypass the so-called swampland conjectures and, in particular, the trans-Planckian censorship conjecture.This turns out to be yet another implication of being able to interpret the fluctuations ontop of de Sitter as a state built out of the underlying Minkowski vacuum, namely theAgarwal-Tara state. More interestingly, this gives us a simple way to argue against theage-old instabilities of de Sitter spacetime against radiative corrections due to the choice ofthe vacuum associated with the mode functions for de Sitter space. During the temporalregime for which our solution is under quantitative control, these instabilities never show– 113 –p as the artifacts of the time-dependent frequencies get explained as mentioned above.Finally, we find the remarkable result that the interpretation of de Sitter as a Glauber-Sudarshan state also helps in the microscopic understanding of the entropy associated withthe de Sitter horizon. This is so because the modes which are responsible for the creationof the coherent state are themselves part of a highly interacting theory and are, therefore,necessarily entangled between themselves. Any entanglement between quantum modesmust result in a nontrivial von Neumann entropy, which we reinterpret as the entropyassociated with the resulting de Sitter state. The crucial role of the interactions can beeasily understood on taking the limit in which they go to zero: Such a limit not onlyresults in the entanglement entropy going to infinity but also ensures there is no longer aGlauber-Sudrashan state any longer, thus reducing the resulting space-time to the warped-Minkowski one. A detailed calculation for this entanglement entropy and its relation tothe famous semiclassical result of the one-quarter horizon area is left for future work andwe only show how it is going to be finite in our case. Of course, the calculation of theentanglement entropy needs to be done using perturbation theory, to evaluate the matrixelements, and is thus, in principle, able to extend beyond the semiclassical result, as itshould be for the calculation on entropy of de Sitter from a microscopic theory.In retrospect our identification of de Sitter space to the Glauber-Sudarshan state in fullstring theory should not come as a big surprise, as a familiar lore in string theory identifies every curved background as some condensates of gravitons by exponentiating the vertexoperator, much like the way discussed in footnote 34. The special case here is, because ofthe temporal dependence of the metric components, the Glauber-Sudarshan state could bedefined on-shell, at least to a large extent. For generic curved background, with or withouttime-dependences, this may not always be possible and the Glauber-Suarshan state shouldbe defined off-shell . This off-shell formalism of the coherent states is in concordance withthe expectation from string field theory where similar constructions show up, althoughthe analysis gets technically challenging. Thus instead of going far off-shell, it will beinteresting to ask if such on-shell description can still be given for the inflationary modelsin string theory [90] as they are close to the de Sitter space that we discussed here. Webelieve this is possible, and more details will be presented in near future. Acknowledgements:
We would like to thank Robert Brandenberger, Maxim Emelin, Shahin Sheikh-Jabbari andSav Sethi for useful discussions. The work of KD is supported in part by the NaturalScience and Engineering Research Council of Canada (NSERC). SB is supported in partby funds from NSERC, from the Canada Research Chair program, by a McGill SpaceInstitute fellowship and by a generous gift from John Greig. Even the simpler case of a Higgs vacuum cannot be defined as an on-shell Glauber-Sudarshan state. Ad-ditionally, as we saw in the decomposition (2.17), there are small off-shell pieces from the time-independentparts of (2.1). Once the system becomes completely off-shell a decomposition like (2.17) could still be used,but now it’s only the second part of (2.17) that would be relevant. See also [89]. – 114 – eferences [1] G. W. Gibbons, “Thoughts on tachyon cosmology,” Class. Quant. Grav. , S321 (2003)[hep-th/0301117]; “Aspects Of Supergravity Theories,” Print-85-0061 (CAMBRIDGE).[2] J. M. Maldacena and C. Nunez, “Supergravity description of field theories on curvedmanifolds and a no go theorem,” Int. J. Mod. Phys. A , 822 (2001) [hep-th/0007018].[3] K. Dasgupta, R. Gwyn, E. McDonough, M. Mia and R. Tatar, “de Sitter Vacua in Type IIBString Theory: Classical Solutions and Quantum Corrections,” JHEP , 054 (2014)[arXiv:1402.5112 [hep-th]].[4] K. Dasgupta, M. Emelin, E. McDonough and R. Tatar, “Quantum Corrections and the deSitter Swampland Conjecture,” JHEP , 145 (2019) [arXiv:1808.07498 [hep-th]].[5] S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, “de Sitter vacua in string theory,”Phys. Rev. D , 046005 (2003) [arXiv:hep-th/0301240 [hep-th]].[6] I. Bena, M. Grana, S. Kuperstein and S. Massai, “Anti-D3 Branes: Singular to the bitterend,” Phys. Rev. D , no. 10, 106010 (2013) [arXiv:1206.6369 [hep-th]];“Polchinski-Strassler does not uplift Klebanov-Strassler,” JHEP , 142 (2013)[arXiv:1212.4828 [hep-th]].[7] J. Moritz, A. Retolaza and A. Westphal, “Toward de Sitter space from ten dimensions,”Phys. Rev. D , no. 4, 046010 (2018) [arXiv:1707.08678 [hep-th]];“On uplifts by warped anti-D3 branes,” Fortsch. Phys. , no. 1-2, 1800098 (2019)[arXiv:1809.06618 [hep-th]];F. Carta, J. Moritz and A. Westphal, “Gaugino condensation and small uplifts in KKLT,”arXiv:1902.01412 [hep-th].[8] S. Sethi, “Supersymmetry Breaking by Fluxes,” JHEP , 022 (2018) [arXiv:1709.03554[hep-th]].[9] E. A. Bergshoeff, K. Dasgupta, R. Kallosh, A. Van Proeyen and T. Wrase, “D3 and dS,”JHEP , 058 (2015) [arXiv:1502.07627 [hep-th]];E. McDonough and M. Scalisi, “Inflation from Nilpotent K¨ahler Corrections,” JCAP ,no. 11, 028 (2016) [arXiv:1609.00364 [hep-th]];R. Kallosh, A. Linde, E. McDonough and M. Scalisi, “de Sitter Vacua with a NilpotentSuperfield,” Fortsch. Phys. , no. 1-2, 1800068 (2019) [arXiv:1808.09428 [hep-th]];“4D models of de Sitter uplift,” Phys. Rev. D , no. 4, 046006 (2019) [arXiv:1809.09018[hep-th]];“dS Vacua and the Swampland,” JHEP , 134 (2019) [arXiv:1901.02022 [hep-th]];S. Kachru, M. Kim, L. McAllister and M. Zimet, “de Sitter Vacua from Ten Dimensions,”arXiv:1908.04788 [hep-th].[10] Y. Hamada, A. Hebecker, G. Shiu and P. Soler, “On brane gaugino condensates in 10d,”JHEP , 008 (2019) [arXiv:1812.06097 [hep-th]];Y. Hamada, A. Hebecker, G. Shiu and P. Soler, “Understanding KKLT from a 10dperspective,” JHEP , 019 (2019) [arXiv:1902.01410 [hep-th]].[11] N. Cribiori, R. Kallosh, C. Roupec and T. Wrase, “Uplifting Anti-D6-brane,”arXiv:1909.08629 [hep-th];R. Kallosh and A. Linde, “Mass Production of Type IIA dS Vacua,” arXiv:1910.08217[hep-th]. – 115 –
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